Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 1 1.1 -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18, 21, 24, 30,...} Then A B= and A B= Sets of Numbers Name(s) for the set 1, 2, 3, 4, Natural Numbers Positive Integers Symbol(s) for the set, -3, -2, -1 Negative integers 0, 1, 2, 3, 4, Non-negative Integers Whole Numbers, -3, -2, -1, 0, 1, 2, 3, Integers Note: is the German word for number.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 2 4 5, 17 13, 1 29 Rational Numbers Note: This is the first letter of 1. Circle One: a) 5 is a rational number!!! TRUE!! FALSE b) π 4 is a rational number!!! TRUE!! FALSE c) 16 5 is a rational number!! TRUE!! FALSE d) 17 5 is a rational number!! TRUE!! FALSE e) 0 is a rational number!!! TRUE!! FALSE Definition of a rational number: If a, b! and b 0, then a b!. Translation: Decimal Expansions of Rational Numbers: Fact: The decimal expansions of rational numbers either or Examples! 1 4 = ( This is a terminating decimal.) 4 9 = (This is a repeating decimal.)
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 3 You know how to express a terminating decimal as a fraction. For example.25 = 25, but how 100 do you express a repeating decimal as a fraction? 2. Express these repeating decimals as the quotient of two integers: a).373737 or.37 Let x =.37373737... Then 100x = b).123 Let x =.123123123... Then There are a lot of rational numbers, but there are even more irrational numbers. Irrational numbers cannot be expressed as the quotient of two integers. Their decimal expansions never terminate nor do they repeat. Examples of irrational numbers are The union of the rational numbers and the irrational numbers is called the or simply the. It is denoted by.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 4 4. What properties are being used? Name of Property a) ( 40) + 8(x + 5) = 8(x + 5) + ( 40) a + b = b + a a,b! b) 8(x + 5) + ( 40) = 8x + 40 + ( 40) a(b + c) = ab + ac a,b,c! c) 8x + 40 + ( 40) = 8x + (40 + ( 40)) (a + b) + c = a + (b + c) a,b,c! d) 0 + 5 = 5 + 0 = 5 0 + a = a + 0 = a a! e) 7 + ( 7) = ( 7) + 7 = 0 For each a!, another element of! denoted by a such that a + ( a) = ( a) + a = 0 f) 6 1 = 1 6 = 6 1 a = a 1 = a a! For each a!, a 0 another g) 4 1 4 = 1 4 4 = 1 element of! denoted by 1 a a 1 a = 1 a a = 1 such that
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 5 Properties of Real Numbers Notice that the commutative property considers the of the numbers. When demonstrating the associative property, the order of the numbers stays the same but the changes. When do we use the commutative property of addition? Consider 7 + 28 + 43.! The rules for order of operation tell us to add left to right, but the commutative property of addition allows us to change the order to make our life easier: 7 + 29 + 43 = When do we use the associative property of multiplication? Consider 16 25. 16 25 = Is subtraction commutative? Example: Is division commutative? Example: Consider ( 12 3) 4 = while 12 ( 3 4) = So is subtraction associative? Consider ( 12 3) 4 = while 12 ( 3 4) = So is division associative?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 6 Notice that there are 2 identities: The additive identity is.! The multiplicative identity is. Notice there are 2 kinds of inverses. The additive inverse of 4 is. Note that The additive inverse of -9 is. Note that What is true about the sum of a number and its additive inverse? Every real number has an additive inverse. In rigorous settings, subtraction is defined as addition with the additive inverse. In other words 5 2 = 5 + ( 2) or more generally a b = a + ( b) The multiplicative inverse of 4 is. Note that The multiplicative inverse of -9 is Note that What is true about the product of a number and its multiplicative inverse? Almost all of the real numbers have a multiplicative inverse. Which real number does NOT have a multiplicative inverse?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 7 Absolute Value 4 = 9 = Formal definition for absolute value!!!! x = It will help if you learn to think of absolute value as (Notice that -4 is 4 units away from the origin.) Distance between two points: 3.) Find the distance between a) 4 and 1 b) -1 and 5 c) 4 and -2 d) -3 and -4 e) 1 and x f) x and y
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 8 Notice that! 4 1 = 1 4! and! 3 2 = 2 ( 3).!!! In general x y = Notice that! 5 + 4 =! whereas 5 + 4 =!! So, in general x + y A solution set is the set of values that make a (mathematical) statement true. Interval Notation Determine the solution sets for the following inequalities Number Line Inequality Interval Notation x 3 x > 3 x > 3 x < 3 This kind of bracket indicates that the endpoint of the interval is included. This kind of parenthesis means that the endpoint of the interval is not included. The interval notation for this set 3 x < 3 is!!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 9 Understanding Zero True or False :!! True or False :!! True or False :!! 0 6 = 0! 6 0 = 0!! 0 0 = 0!! For a, b!, a = 0 and b Another way to say this: a fraction is zero when its is zero and its is not. This concept will be very important when we solve rational equations like 1 x +1 = x + 2 3 x Notation inconsistency: When we write 4x it means the product of 4 and x. When we write 4 1 3 it means the sum of 4 and 1 3 4 1 3 is usually called a mixed number whereas the corresponding is termed an improper fraction. There is nothing improper about an improper fraction.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.1! Page 10 Extra Problems for 1.1 -- Real Numbers 1. Below is a sequence of algebraic statements. a b Please indicate which of the properties is being used. 1 1 a) (3b + 3a) = (3a + 3b)! a + b a + b 1 1 b) (3a + 3b) = 3(a + b)! a + b a + b 1 1 c) 3(a + b) = 3 (a + b)! a + b a + b 1 d) 3 (a + b) = 3(1)!!! a + b e) 3(1) = 3!!!!! 2. a) All real numbers have a multiplicative inverse. TRUE FALSE b) x y = y x x, y! TRUE FALSE c) 1! TRUE FALSE d) 1! TRUE FALSE e) 0! TRUE FALSE f) 0.1763! TRUE FALSE g) π 2! TRUE FALSE h) 25! TRUE FALSE
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 11 4.4 -- Linear Equations in Two Variables A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3, -1), (0, 5), (1, -3), (2, 3)} This is a set of ordered pairs of real numbers, so it is a. Each ordered pair of real numbers corresponds to a on the Cartesian plane. The set of all points corresponding to a relation is the of the relation. Graph the relation given above. The of a relation is the set of all first elements of the ordered pairs. The of a relation is the set of all second elements of the ordered pairs. What is the domain of this relation?! What is the range of this relation?!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 12 A relation can be infinite: Consider {(x, y) -3 x < 2, 1< y 4} List 3 different elements of that set:!!!! Graph the relation What is the domain of this relation? What is the range of this relation? Equations can be used to define relations Consider the set {(x, y) x + y = 3} List three elements of that set:!!!! It is important to note that the elements of this relation are the solutions to the equation. Graph the relation What is the domain of this relation? What is the range of this relation?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 13 Consider the set {(x, y) x = y } 2 List several elements of that set. It is important to note that the elements of this relation are the solutions to the equation. What is the domain of this relation? What is the range of this relation? Consider the set {(x, y) y = x } 2 List several elements of that set. It is important to note that the elements of this relation are the solutions to the equation. What is the domain of this relation? What is the range of this relation?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 14 Functions True or False: A function is a relation in which no two different ordered pairs have the same first element. True or False:! A function is a relation where there is only one output (called the y-value), for each input (called the x-value). True or False:! A function is a rule that assigns exactly one element in a set B (called the range) to each element in a set A (called the domain). Which of the following represent a function? {(1, 2), (1, 3), (2, 3)} This set of ordered pairs is {(1, 3), (2, 3), (3, 3), (4, 3)} This set of ordered pairs is Functions as a Rule: 1. If f (x) = 3x + 4 then f (2) = 2. If f (x) = 3x 2 + 4x +1 then f ( 1) = Note: Order of operations dictates that 3 2 = while ( 3) 2 = These rules create sets of ordered pairs of real numbers ( x, f (x)) In these cases the domain is the set of numbers for which f (x) is defined. The range is the set of values that f (x) attains.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 15 The set of all points corresponding to a function is the of the function. A curve in the xy plane is the graph of a function iff no line the curve So if there is a line that the curve then the curve is the graph of a This is called {(x, y) x = y } 2 Is a relation, but It the vertical line test. {(x, y) y = x } 2 Is a relation and a It the vertical line test. So what we are seeing is that there are 2 common ways to define a function. A function can be defined with an equation such as {(x, y) y = x } 2 or a function can be defined with a rule such as {(x, f (x)) f (x) = x } 2 But having seen all of this, the truth is that we seldom use the set notation, just the equation or the f (x) notation.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 16 Linear Equations Circle the equations below if it is an example of a linear equation in 2 variables. 3x + 4y = 5!! y = x 9 πx + 4y = 5 x + y + z = 0 3x + xy + 4y = 5 x + 4y = 2 x 2 + 4y = 2 y = 2 x v = 32t +16 y = 2 x = 5 x = y Any equation that can be written in the form Ax + By + C = 0 where A and B are not both is called a The set of ordered pairs of real numbers {(x, y) x = 3} is a relation, but it is a
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 17 You already know a lot about lines, but let s review quickly: Consider x + y = 4 x y. When you create a table to list points to create the graph, you are determining the solutions to the equation. Solutions to linear equations in two variables are ordered pairs of real numbers that when substituted into the equation create an identity. The coordinates of all of the points on the line are solutions to the equation. How many solutions exist for this equation? Graphing lines with both intercepts Consider x + 3y = 3 When x = 0, y = and when y = 0, x =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 18 Intercepts (crossings) The y -intercept is the point where the graph crosses the It occurs when The x -intercept is the point where the graph crosses the It occurs when Consider 2x + y = 4. Solve this equation for y. What is the y -intercept?! When x = 0, y = The point is What is the slope of this line? Slope Calculate the slope m of the line on the graph below.! m = Δy Δx =! Note Δ is the Greek letter.! Δ or D is for! Some people like the phrase!! slope = What is the equation of this line? This y = mx + b form of a linear equation is known as the form.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 19 Thinking about slope as m = Δy Δx... Note that for horizontal lines, Δy =, so the slope of a horizontal line is Note that for vertical lines, Δx =, The slope of a vertical line is Parallel lines do not. Parallel lines have the same. When 2 lines are perpendicular their slopes are. If the slope of the first line is m 1 = 5, then the slope of a second line perpendicular to the first line would be m 2 = Determine the equation of the line through (-1, -2) that is perpendicular to y = 3x + 4.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 20 Determine the equation of the line through (2, 0) and (-4, 3). The Midpoint of a Line Segment. Consider the points A (-3, 0) and B ( 5, -4). Find the midpoint of the segment connecting these 2 points.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 21 Remember: The number half between a and b is the average of a and b Given a line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ), the midpoint of that segment is Consider a line segment through (2, 0) and (-4, 3). What are the characteristics of a perpendicular bisector of a line segment?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 4.4! Page 22 Extra Problems for 4.4 -- Linear Equations in Two Variables 1. Determine the equation of the line parallel to x + 3y 6 = 0 that passes through the origin. 2. What is the slope-intercept form of the perpendicular bisector of the line segment with endpoints (-6, -4) and (-10, 2)
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 23 8.2 Algebraic Methods of Solution (Systems of Linear Equations) The set {(x, y) x + y = 3} is a set of ordered pairs of real numbers. It is a function, and all of the elements of this set are solutions to the equation x + y = 3 Definition: A solution to an equation in 2 variables is an ordered pair of real numbers (x, y) that, when substituted into the equation, make the equation an identity. (In math an identity is a true statement.) 1. a) List 3 examples of solutions for the equation x + y = 3.!! b) How many solutions exist for the equation x + y = 3? c) Graph all of the solutions of x + y = 3 2. a) List 3 examples of solutions for the equation x y = 1. b) How many solutions exist for the equation x y = 1? c) Graph all of the solutions of x y = 1 Do these two solution sets have any common elements? A solution to a system of equations in 2 variables is an ordered pair of real numbers (x, y) that, when substituted into all of the equations in the system, make all of the equations identities.!! is the solution to the system x + y = 3 x y = 1 because and
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 24 2. How many solutions exist for the following system of equations?! x + y = 1 x + y = 2!!!! x + y = 1 x + y = 1!!!x + y = 1 3x + 3y = 3 Geometrically there are 3 possibilities for the graphs of 2 lines. Finding solutions to systems of linear equations (when they exist!) By Graphing! Solutions to systems of equations correspond to the intersections of the graphs of the equations. By Substitution 3. Find all of the solutions, if any exist, to the system x + 3y = 7 2x y = 7 Solve the first equation for x: Substitute that x into the second equation: Solve the second equation for y: Substitute that value of y into either equation to find x The solution of this system is the point
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 25 Note that when substituting the solution into each equation, an identity (a true statement) is formed. It checks. Also, the solution is the point on the graph where the 2 lines intersect. By Elimination Our goal will be to add or subtract the equations in a way that eliminates one of the variables. Part of the process will remind you of finding a common denominator. Why is elimination a legitimate technique? 2x + y = 1 x y = 5 Start with the first equation!!!!!! 2x + y = 1 Rewrite the equation adding 5 to both sides Substitute x y for the 5 on the left hand side. In essence we have added the 2 equations to one another. 4. Solve 2x + 3y = 4 5x + 6y = 7 5. Solve 6x + 7y = 18 4x + 3y = 12
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 26 Applications 6. An airplane makes the 2400 miles trip from Washington D.C. to San Francisco in 7.5 hours and makes the return trip in 6 hours. Assuming that the plane travels at a constant airspeed and that the wind blows at a constant rate from west to east, find the plane s airspeed and the wind rate.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 27 7. To raise funds, the hiking club wants to make and sell trail mix. Their plan is to mix dried fruit worth $1.60 per pound with nuts worth $2.45 per pound to make 17 pounds of a mixture worth $2 per pound. How many pounds of dried fruit and how many pounds of nuts should they use?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 28 8. How many gallons of each of a 60% acid solution and an 80% acid solution must be mixed to produce 50 gallons of a 74% acid solution?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 8.2! Page 29 Extra Problems 8.2 Algebraic Methods of Solution 1. Demonstrate your skill with the elimination method to solve 9x + 8y = 7 12x +10y = 9 2. A lab assistant needs 30 milliliters of a 50% methyl alcohol solution. She has access to a 70% solution and a 40% solution. How many milliliters of the 70% solution should she use?
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.2! Page 30 5.2 Functions (Graphs of Basic Functions) Graph the following library of basic functions. It is important to be able to recognize and sketch these graphs with ease! The constant function f (x) = c c is a real number Domain Range Identity function f (x) = x Domain Range Squaring Function f (x) = x 2 Cubing Function f (x) = x 3 Domain Range Domain Range
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.2! Page 31 Square Root Function f (x) = x Domain Range Absolute Value Function f (x) = x Domain Range Reciprocal Function f (x) = 1 x Cube Root Function f (x) = 3 x Domain Range Domain Range 3 3 27 =! 8 =! 3 3 3 1 =! 1 = 8 =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.2! Page 32 Piecewise Functions x, x < 0 1. Let f (x) = 3, x = 0 x, x > 0 a) f ( 2) = b) f (4) = c) Graph f (x) d) x intercept e) y -intercept f) domain g) range Assume that the graph below is that of the function g(x). Evaluate g( 1) g 3 2 g( 3) Write a definition for g(x) Domain: Range:
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.2! Page 33 Consider the graph on the left. Examine how the graph is changing as you look at it from left to right. On the left side, the graph is going down. That is the interval where the function is decreasing. At x = 1 2 there is a change. The graph starts to go up. This is the interval where the function is increasing. Increasing, Decreasing, and Constant A function f (x) is increasing on an interval I iff x 1 < x 2 x 1, x 2 I A function f (x) is decreasing on an interval I iff x 1 < x 2 x 1, x 2 I A function f (x) is constant on an interval I iff x 1 < x 2 x 1, x 2 I The graph below is associated with a function f (x) Domain Range x intercept y -intercept f (x) is increasing on f (x) is decreasing on f (x) is constant on
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.2! Page 34 Extra Problems for 5.2 Functions (Graphs of Basic Functions) x + 7, x < 2 1. If f (x) = (x + 2) 2 + 5, -2 x 1 6, 1 < x, then a) f ( 3) = b) f (π ) = 2. Given the graph of the function below, determine the following (If there are none, write NONE.) a) domain b) range c) interval(s) on which the function is decreasing d) interval(s) on which the function is increasing e) maximum value of the function e) minimum value of the function
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 35 9.1 Angles and Circles From now on angles will be drawn with their vertex at the The angle s initial ray will be along the positive. Think of the angle s terminal ray as starting along the positive x-axis, and then swinging into its position. If the terminal ray swung away from the x-axis in a counterclockwise direction, then the angle has measure. If the terminal ray swung away from the x-axis in a clockwise direction, then the angle has measure. Radians An angle with its vertex at the center of circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. Approximate: 1 radian 57.3! Exact: π radians = 180! This is fun to watch: https://en.wikipedia.org/wiki/radian#/media/file:circle_radians.gif
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 36 The circle below has a radius of 1 unit. It is called the. The circumference of a unit circle is If a terminal ray swings through an entire rotation, you would say it has a measure of You could also say that it has a measure of.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 37 1. Sketch the following angles on the unit circle below a) π 4 b) 2π 3 c) 7π 6 d) 3π 2 e) 7π 4 Conversions: Since π radians =, so 1 = = 2. Convert a) 45! = b) 5π 3 radians = Since 1 revolution = 1 = = Since 1 revolution = 1 = =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 38 Convert to revolutions (or back to degrees and/or radians) a) 450! = b) 15π 4 radians = c) 3 1 3 revolutions = Sketch θ 1 = 3π and θ 2 = π on the same unit circle 3π and π are called because they share the same To find an angle that is coterminal to θ, just add or subtract Another way to say this: To find an angle that is coterminal to θ, just add or subtract List 2 other angles that are coterminal angles with π 2 List 2 other angles that are coterminal angles with 2π 3
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 39 An angle is called acute if its measure is between An angle is called obtuse if its measure is between Two angles are called complementary if the sum of their measures is An example of complementary angles: θ 1 = and θ 2 = Two angles are called supplementary if the sum of their measures is An example of supplementary angles: θ 1 = and θ 2 = A line which intersects the circle twice is called a A line which intersects the circle at exactly one point is called a The region inside of a circle is called a Any piece of the circle between two points on the circle is called an Any line segment between 2 points on the circle is called a Any piece of the disk between 2 radial lines is called a An angle whose vertex is at the center of a circle is called
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 40 Arc Length: (Think about the fraction of the circumference.) θ = 120! r = 6 units θ = π 4 r = 5 units Area of a sector: (Think about the fraction of the area of the circle.) θ = 30!, r = 2 units θ = 7π 6 r = 3 units
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 41 When using degrees to measure the central angle! Area of a sector =!!!!!! Length of arc =! When using radians to measure the central angle! Area of a sector =!!!!!! Length of arc =!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.1! Page 42 Extra Problems for 9.1 Angles and Circles 1. If a circle has a 10 meter diameter, find the exact area of the sector subtended by a central angle of 18 degrees. 2. If a circle has diameter 12 cm, determine the arc length of the sector subtended by a central angle of 40!. 3. a) 800! = radians b) 800! = revolutions 4. True or False: α = π 6 and β = 25π 6 are coterminal angles.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.5! Page 43 5.5 - Combinations of Functions The basic functions of 5.2 can be combined. Let f (x) = 2, and g(x) = x, and h(x) = 1 x a) What is the domain of f (x)? b) What is the domain of g(x)? c) What is the domain of h(x)? d) ( f + g)(x) = e) What is the domain of ( f + g)(x)? f) ( h f )(x) = g) What is the domain of ( h f )(x)? h) ( fg)(x) = i) What is the domain of ( fg)(x)? j) f g (x) = k) What is the domain of f g (x)? Let f (x) be a function with domain A and let g(x) be a function with domain B. Then the domains of ( f + g)(x), ( f g)(x), and ( fg)(x) are alla B. the domain of f g (x) is { x A B, g(x) 0 }.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.5! Page 44 Example 1. f (x) = 4 x and g(x) = x + 5 The domain of f (x) is A = The domain of g(x) is B = ( f + g)(x) = The domain of ( f + g)(x) is f g (x) = The domain of f g (x) is Example 2. f (x) = x and g(x) = 1 x then ( fg)(x) = the domain of f (x) is the domain of g(x) is so the domain of ( fg)(x) is In other words ( fg)(x) = is undefined at so its graph would look like:
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.5! Page 45 Function Composition Definition For functions f (x) and g(x), the composition of functions ( f! g)(x) is defined as 2. Let f (x) = x + 5 and g(x) = 1 x a) ( f! g)( 4) = b) (g! f )( 4) = c) ( f! g)(x) = d) (g! f )(x) = In general ( f! g) ( x) =. The domain of ( f! g) ( x) is the set of allx in the domain of g(x) such that g(x) is in the domain of f (x). ( f! g) ( x) = is defined whenever both g(x) and f (g(x)) are defined. Practicing with function notation: 2. Let f (x) = x 2 + x Determine a) f (2) b) f ( 3) c) f ( ) d) f (x 2 ) e) f (4x) f) f (x + h)
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.5! Page 46 3. f (x) = 4 x and g(x) = x + 5 ( f! g) ( x) = Domain of ( f! g) ( x) is Example 2. If ( f! g) ( x) = x + 4 3 x + 4, then f (x) = and g(x) = When functions are combined by adding, subtracting, multiplying, dividing, and/or composition, the result often needs to be simplified. Moreover, it will be important to examine the domain of the resulting function. That work will be done largely in sections 1.2, 5.3 and 5.4 4. Here are the graphs of two functions. If the line is the graph of f (x) and the parabola is the graph of g(x), determine a) ( f! g) ( 4) b) ( g! f )( 4) c) ( f! g) ( 0) d) ( g! f )( 0) e) ( f + g) ( 0) ( f + g) ( 4)
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.5! Page 47 Extra Problems for 5.5 - Combinations of Functions 1. a) If f ( x) = x 2 + 3x 5, simplify f ( x + h) = b) If g( x) = x + 4 x, simplify f ( x + h) = 2. f (x) = 3x 2 + 4, g( x) = 5x + 6 a) Simplify ( f! g) ( x) = b) Simplify ( g! f )( x) =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 48 1.2 - Exponents and Radicals Multiplication is shorthand notation for repeated addition 4 + 4 + 4 + 4 + 4 = 5 4 Exponents are shorthand notation for repeated multiplication. So 2 2 2 2 2 = xixixix = You probably know that 3 1 = and x 5 = this is because by definition x 1 =. In general if m Ν, then x m = Properties of exponents Example In general x 4 x 3 x 8 x 5 x m x n x m x n x 3 x 3 x 0 for x 0 Note: 0 0 0 and 0 0 1 0 0 is an More Properties of Exponents Example In general ( x 2 ) 3 (xy) 3 (xy) m x 2 y x m y
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 49 1. Simplify a) ( 8) 2 b) 8 2 c) 3 0 d) 1 2 6 e) 1 x 2! e) x 3 4 f) 3 x 1 g) 25x 5 y 2 15x 2 y 1 3 h) 12x 3 y 2 4x 4 y 1 2
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 50 Roots or Radicals TRUE or FALSE: 4 = ±2. Important distinction: x 2 = 25 has solutions: x = and x = 25 represents exactly number. 25 =. Recall that 2 5 = 32. Now suppose you have x 5 = 32 with the instructions: Solve for x. How do you express x in terms of 32? The vocabulary word is In this case x equals the. The notation is x = or x = This second notation is called a fractional exponent. ( ) 1 4 4 Examples:! 81 = 81 = because 3 =. ( ) 1 3!!! 125 = because 5 =. Properties of Roots (just like properties of exponents!) Example In general 3 27i8 m xy 3 27 3 8
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 51 Example In general 36 4 36 4 m m x y 2 3 Now consider 8. Notice that ( 8 2 ) 1 3 = Also 8 1 3 2 = 2 3 So it would seem that 8 = In fact, more generally, it is true that x m n = 3 Similarly, consider 64 = 64 1 2 1 3 = 3 Whereas, 64 = 64 1 3 1 2 = 3 So it would seem that 64 In general m n x
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 52 Simplify n x n for all n!. On Worksheet 1 you graphed y = 3 x 3. Hopefully you found that its graph looked just like the graph of y = x. In other words, you saw that 3 x 3 = You also graphed y = 2 x 2. Hopefully you found that its graph looked just like the graph of y = x. In other words, you saw that 2 x 2 = Most people believe that 5 2 5 = and agree easily that n 2 n = n But ( 2) 4 ( ) 5 4 5 = while 2 = So hopefully you see believe n ( 2) n = 2 whenever n is and n ( 2) n = 2 whenever n is So to simplify n x n for any x! and any n! we have n x n = When simplifying, we don t want to use absolute value signs if they are unnecessary. Obviously we would write 4 = 4. But also x 4 = Also, for the time being, if n is even, n x is defined only if x 0. In other words, for the time 4 being, 4 and 17 are not defined. So x 3 is only defined if x 0. That means that x 3 =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 53 Simplifying Radicals A radical expression is simplified when the following conditions hold: 1. All possible factors ( perfect roots ) have been removed from the radical. 2. The index of the radical is as small as possible. 3. No radicals appear in the denominator. 2. Simplify 5 a) 32 b) ( 216) 1 3 c) 9 1 2 d) 64 6 e) ( 13) 6 5 f) ( 13) 5 g) 4 16x 4 h) 3 x 5 8 i) 16 j) 125 64 2 3
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 54 Rationalizing the Denominator Rationalizing the denominator is the term given to the techniques used for eliminating radicals from the denominator of an expression without changing the value of the expression. It involves multiplying the expression by a 1 in a helpful form. 3. Simplify a) 1 2!!!!! b) 1 3 2 c) 3 1 4x!!!!!! d) 4 16 27x Notice: ( x + 3) ( x 3) =. To simplify 1 2 5 we multiply it by 1 in the form of
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 55 So 1 2 5 = and 9 x + 3 5 =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 56 Adding and Subtracting Radical Expressions Terms must be alike to combine them with addition or subtraction. Radical terms are alike if they have the same index and the same radicand. (The radicand is the expression under the radical sign.) 4. Simplify a) 5 + 2 7 3 5 7 b) 3 128x 4 3 2x 4 Please notice 9 + 16 =! Whereas 25 = In other words In general True or False: 2x + 3x = 5x True or False: 2x + 2x = 4x 2x + 2x =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 57 Interesting cases of n x n when n is even For the time being if n is even, n x is undefined if x < 0 in other words 6 2 is undefined. Moreover, we want to remember that when x > 0, x = x ; we always simplify 4 = To simplify the following, start by considering for what values of x the expression is defined 4 x 4 Here the index of the radical is even, but the expression under the radical is always positive, so the expression is defined for all values of x 4 x 4 = 4 x 5 Here the index of the radical is even, but x 5 could be negative if x was negative. We have to recognize that 4 x 5 is only defined if x > 0 so 4 x 5 = 4 x 8 Here the index of the radical is even, and the expression under the radical is always positive, so the expression is defined for all values of x 4 x 8 = 4 x 12 Here the index of the radical is even, and the expression under the radical is always positive, so the expression is defined for all values of x 4 x 12 = Life is so much easier when the index of the radical is odd, because when the index of the radical is odd, the expression is always defined 3 x 3 5 5 =!! 1 =!! 32 =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 58 Extra Problems for 1.2 - Exponents and Radicals 1. a) 3 3 xy = x 3 ( )( y ) x, y! TRUE FALSE b) The multiplicative inverse of 2.is 2 2 TRUE FALSE 2. Simplify completely a) 1 2 1 2 = b) 1 3 16 =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.2! Page 59 3 c) 63 + 28 56 = d) 64x 8 y 12 4 = e) 32x 4 y 4 243x 6 y 3 5 =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 60 5.3 -- Transformations of Functions Vertical Shifts x f (x) = x g(x) = x + 2 h(x) = x 1-3 3-2 2-1 1 0 0 1 1 2 2 3 3 These three absolute value functions are defined everywhere. For each the domain is Range of g(x) Range of h(x) x intercept for g(x) x intercept for h(x) g(x) is increasing on h(x) is decreasing on In general, adding a constant to a function shifts the graph 1. Write a function g(x) that shifts the graph of f (x) = x 3 4 units down. 2. Write a function h(x) that shifts the graph of f (x) = x 3 units up.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 61 Horizontal Shifts x -5-4 -3-2 -1 0 1 2 3 4 5 6 f (x) = x 2 9 4 1 0 1 4 9 g(x) = ( x 3) 2 h(x) = ( x + 2) 2 Domain of f (x), g(x), h(x) Range of f (x), g(x), h(x) vertex of g(x) vertex of h(x) x intercept for g(x) x intercept for h(x) y -intercept for g(x) y -intercept for h(x) g(x) is increasing on h(x) is decreasing on
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 62 x -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 f (x) = x 0 1 2 3 g(x) = x + 4 h(x) = x 1 For square root functions, there is a point where the x value makes the radicand equal 0. We will call this the (The radicand is the expression under the radical sign.) anchor point for g(x) anchor point for h(x) domain of g(x) domain of h(x) x intercept for g(x) x intercept for h(x) g(x) is increasing on h(x) is decreasing on Range of f (x), g(x), h(x)
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 63 In general adding a constant to x before applying the function results in a shift of the graph. 3. f (x) = x Write the function g(x) that shifts the graph of f (x) four units to the right. g(x) = Domain of g(x) The anchor point for g(x) 4. Write the function h(x) that shifts the graph of f (x) = x two units to the left. h(x) = vertex of h(x) = x intercept for h(x) y -intercept for h(x)
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 64 Reflections Let f (x) = x 2!!! Let g(x) = f (x) = x 2 Range of f (x)!!! Range of g(x) x f (x) = x 2 g(x) = x 2-3 9-2 4-1 1 0 0 1 1 2 4 3 9 We can see f (x) is a of f (x) through the Evaluating f (-x) If f (x) = x 2 then f ( x) =!! If g(x) = 1 x then g( x) = If h(x) = x 3 then h( x) =!!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 65 If r(x) = x then r( x)= Let p(x) = r( x) = What is the domain of p(x)? Graph r(x) = x and p(x) = x Consider f (x) = (x 3) 2 Let g(x) = f ( x) = Graph f (x) and g(x). Here we see that f ( x) is a of f (x) through the
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 66 Stretching and Shrinking x f (x) = x 3 g(x) = 1 2 1 4 x 3-2 -8-1 -1 0 0 1 1 2 8
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 67 If c > 1, then the transformation y = cf (x) f (x) by a factor of c. Whereas if 0 < c < 1, then the transformation y = cf (x) f (x) by a factor of. Combinations of Transformations: 5. f (x) = x + 4 + 3 a) domain of f (x) b) anchor point c) y intercept To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. d) x intercept! e) range f) f (x) is increasing on! List the transformations in order because the order matters!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 68 6. g(x) = 2 1 x 6 a) domain of g(x) b) anchor point c) y intercept To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. d) x intercept! e) range f) g(x) is increasing on! List the transformations in order because the order matters!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 69 7. h(x) = ( x + 5 ) 1 a) domain of h(x) b) anchor point c) y intercept To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. d) x intercept! e) range f) h(x) is increasing on! List the transformations in order because the order matters!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 70 Determining x-intercepts for transformed absolute value functions 8. g(x) = x +1 + 3 Determine the following: a) vertex b) y -intercept c) another point on the graph from symmetry To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. d) x -intercepts e) range
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 71 9. f (x) = 2 x 3 4 Determine the following: a) vertex b) y -intercept c) another point on the graph from symmetry To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. d) x -intercepts e) range
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 72 Sometimes absolute value functions do not intersect the x-axis. 10. f (x) = x + 2 +1 Determine the following: a) vertex b) y -intercept c) another point on the graph from symmetry To determine the x -intercept we need to determine where That means we want to solve the equation As before we would begin by isolating where the x is hiding. : Quickly we see that this equation does have a. It is always nice when the geometry of a problem agrees with our algebra. d) x -intercepts e) range
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 73 Determining x-intercepts for transformed quadratic functions 11. f (x) = (x 2)2 4 +1 a) domain of f (x) b) vertex c) y -intercept d) symmetric pair to y -intercept To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. e) x intercepts! f) range g) f (x) is increasing on!! h) List the transformations in order because the order matters!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 74 12. g(x) = 2(x +1) 2 5 a) domain of g(x) b) vertex c) y -intercept d) symmetric pair to y -intercept To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. e) x intercepts! f) range g) g(x) is increasing on!! h) List the transformations in order because the order matters!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 75 (x + 3)2 13. h(x) = +1 3 a) domain of h(x) b) vertex c) y -intercept d) symmetric pair to y -intercept To determine the x -intercept we need to determine where That means we want to solve the equation To do this, we isolate where the x is hiding. e) x intercepts! f) range g) h(x) is increasing on!! h) List the transformations in order because the order matters!
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 76 The order of transformations matters! Let the parent function be f (x) = x 1. shift up 3 1. reflect over the x-axis 2 reflect over the x-axis 2. shift up 3 Even Functions: Notice if f (x) = x 2, then f ( x) = In other words f ( x) = A function is iff for every x in the domain of f (x) The graph of an even function has symmetry about the These functions are all even. They all have symmetry about the y-axis. a) f (x) = x 2 b) g(x) = x 2 c) h(x) = x 2 + 3 d) r(x) = 9 x 2
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 77 Odd Functions: Notice if f (x) = x 3, then f ( x) = In other words f ( x) = A function is iff for every x in the domain of f (x). The graph of an odd function has symmetry about the Symmetry about the origin Informally, a graph is symmetric about the origin if when it is rotated 180 about the origin, the graph looks the same. These functions are all odd. They are all symmetric about the origin. a) f (x) = x b) g(x) = x c) h(x) = x 3 d) r(x) = x 3
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 78 To determine if a function f (x) is even or odd or neither, simplify f ( x). If f ( x) = f (x), then f (x) is an function. If f ( x) = f (x), then f (x) is an function. 14. Determine if the following functions are even, odd, or neither a) f (x) = x 2 + 2x +1!!!!!!! b) g(x) = x 3 +1 c) h(x) = 3x 4 x 2 +1!!!!!!! d) q(x) = x2 +1 x 3
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 79 Extra Problems for 5.3 -- Transformations of Functions 1. a) All quadratic functions are even functions. TRUE FALSE b) The graphs of all quadratic functions pass the vertical line test. TRUE FALSE c) The graphs of all quadratic functions intersect the x -axis. TRUE FALSE d) The graphs of all quadratic functions intersect the y -axis. TRUE FALSE e) All even functions are symmetric to the y -axis. TRUE FALSE 2. This is the graph of f (x) on ( 0, ). If f (x) is an odd function, sketch the graph of f (x) on (, 0). Label the coordinates of 3 of the points on the part of the graph that you draw.
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 80 3. Fill in the blanks a) f (x) is an even function if f ( x) = x in the domain of f (x) b) Even functions are symmetric about the c) f (x) is an odd function if f ( x) = x in the domain of f (x) d) Odd functions are symmetric about the 4. This is a graph of the function f (x). Choose the correct definition of f (x) below. a) (x 3) 2 1, x < 1 f (x) = 3, -1 x 1 3 x 1, 1 < x c) (x + 3) 2 1, x < 1 f (x) = 3, -1 x 1 3 1 x, 1 < x e) (x + 3) 2 1, x < 1 f (x) = 3, -1 x 1 3+ x 1, 1 < x b) 1 (x + 3) 2, x < 1 f (x) = 3, -1 x 1 3 1 x, 1 < x d) (x + 3) 2 1, x < 1 f (x) = 3, -1 x 1 3 x 1, 1 < x f) (x 3) 2 1, x < 1 f (x) = 3, -1 x 1 3+ 1 x, 1 < x
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 5.3! Page 81 5. Let f (x) = 4 x and g(x) = x + 5 The domain of f (x) is A = The domain of g(x) is B = In general ( f! g) ( x) =. The domain of ( f! g) ( x) is the set of allx in the domain of g(x) such that g(x) is in the domain of f (x). In other words( f! g) ( x) = is defined whenever both g(x) and f (g(x)) are defined. So for the functions defined above, ( f! g) ( x) = The domain of ( f! g) ( x) is
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.3! 82 1.3 - Complex Numbers 16 exists! So far the largest (most inclusive) number set we have discussed and the one we have the most experience with has been named the real numbers. And x!, x 2 0 But there exists a number (that is not an element of! ) named i and i 2 = Since i 2 =, 1 =, so 16 = i is not used in ordinary life, and humankind existed for 1000 s of years without considering i. i is, however, a legitimate number. i, products of i, and numbers like 2 + i are solutions to many problems in engineering. So it is unfortunate that it was termed imaginary! i and numbers like 4i and 3i are called Numbers like 2 + i and 17 3 5i are called More formally! is the set of all numbers When a complex number has been simplified into this form, it is called a is called the part. b is called the imaginary part. So for the complex number 4 3i, is the real part while is the imaginary part. 1. Put the following complex numbers into standard form: a) ( 16 + 9) + ( 13 100) = b) ( 1+ 2i) (3 4i) = c) ( 5 + 4i)(3 2i) =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.3! 83 d) i 3 = e) i 4 = f) i 5 = Graphing Complex Numbers When we graph elements of!, we use When we graph elements of!, we use the complex plane which will seem a lot like the Cartesian plane. In the Cartesian plane, a point represents In the complex plane, a point represents The horizontal axis is the The vertical axis is the 2. Graph and label the following points on the complex plane: A 3+ 4i!!! B 2 + i C 3 2 3i!!! D 3i E 4!!! F 1 2i
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.3! 84 Absolute Value of a Complex Number: If z! then z is defined as its Calculate 3+ 4i (How far is 3+ 4i from the origin?) 3+ 4i = Consider a + bi, an arbitrary element of!.!! What is a + bi? (How far is a + bi from the origin?) In general z = Notice that z is a real number. It is z s distance from the origin. We did not use i to calculate z. 3. Calculate 5 + i. (How far is 5 + i from the origin?) 5 + i =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.3! 85 Distance Between Two Complex Numbers Plot and label two points in the complex plane z 1 = 3+ 5i and z 2 = 1+ 2i The distance between z 1 and z 2 is Just for fun, calculate z 1 z 2 = And z 1 z 2 = What we have seen is that for this particular z 1 and z 2, the distance between these points is equal to. But this is actually true for all complex numbers. The distance between two general points z 1 and z 2 is Complex Conjugate If z 1 = 3+ 4i, then z 1 =. If z 2 = 1 2i, then z 2 =. Graphically the complex conjugate is the of the number through the More generally if z = a + bi, then z =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.3! 86 4. Put the following complex numbers into standard form. a) 2 3i =!!! b) 4i = c) 5 =!!! d) 3i 2 = Finally notice that ( 3 + 4i) ( 3 4i) = More generally ( a + bi) ( a bi) = in other words ziz = Dividing by complex numbers 5. 25 + 50i 3+ 4i
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 1.3! 87 Extra Problems for 1.3 - Complex Numbers 1. π! TRUE FALSE 2. Calculate, putting complex numbers into standard form. a) ( 5 + 2i)(3 i) = b) 3i + 4 = c) i 4 = 3 If z 1 = 7 6i and z 2 = 4 5i determine z 1 z 2 = 4. ( 3+ i)(5 i) =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.2! 88 9.2 - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of equal measure, then they are. E C A D B Since ABC and ADE are both right triangles sharing the common angle A, they are similar triangles. When 2 triangles are similar, it means the lengths of their corresponding sides are proportional. So AD AB = adjacent leg small adjacent leg big = DE BC opposite leg small = opposite leg big = AE AC hypotenuse small = hypotenuse big If we carefully use algebra to rearrange AD AB = DE BC, we see that BC AB = DE AD or rather that and similarly, since AE AC = DE BC BC then it must be true that AC = DE AE which stated more plainly says
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.2! 89 These rather amazing facts allow us to say that given a right triangle with an angle measured θ, the following ratios are constant (no matter how large or small the triangles are!). sinθ =!!!! cosθ = tanθ =!!!! cotθ = cscθ =!!!!! secθ = Tom s Old Aunt!! Sat On Her!!! Coffin And Howled Notice that sinθ cosθ!!!!!!! In other words sinθ cosθ Similarly cosθ sinθ =!!!! 1 cosθ = In other words 1 cosθ =!!!! Similarly 1 sinθ =
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.2! 90 Consider the equilateral triangle sin π 3 = cos π 3 = tan π 3 = csc π 3 = sec π 3 = π cot = 3 sin π 6 = cos π 6 = tan π 6 = csc π 6 = sec π 6 = π cot = 6
Fry Texas A&M University! Spring 2017! Math 150 Notes! Section 9.2! 91 Consider the isosceles right triangle sin π 4 = cos π 4 = tan π 4 = csc π 4 = sec π 4 = π cot = 4 The trouble with these definitions of our trigonometric functions is that they are defined only for 0 < θ < π. Later we will extend the definitions of these functions for any real value of θ. 2