1.1 Functions. Input (Independent or x) and output (Dependent or y) of a function. Range: Domain: Function Rule. Input. Output.

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1 1.1 Functions Function Function: A rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value uniquely determines one output value. We say the output is a function of the input. Input (Independent or x) and output (Dependent or y) of a function Domain: The set of all inputs Function Rule Range: The set of all outputs Ex: Input Function Rule Output Ordered pair - 0 Double the input Ex. Is {(1,3), (,3), (3,3)} a function? Ex. Is {(3,-1), (3,-3), (3,4)} a function? Ex. If y = x, is y a function of x? Ex. If y = x, is y a function of x? One-to-One Function Sometimes in a relationship each input corresponds to exactly one output, and every output corresponds to exactly one input. We call this kind of relationship a one-to-one function. Ex. Is {(-1,), (,3), (3,4)} a one-to-one function? Ex. Is {(1,-3), (,3), (3,-3)} a one-to-one function?

2 Graph as Functions Ex. Graph the set of points {(1,3), (,3), (3,3)}: Ex. Graph the set of points {(3,-1), (3,-3), (3,4)}: Vertical Line Test The vertical line test is a handy way to think about whether a graph defines the vertical output as a function of the horizontal input. Imagine drawing vertical lines through the graph. If any vertical line would cross the graph more than once, then the graph does not define only one vertical output for each horizontal input. One-to-one function: Ex. Graph the set of points {(-1,), (,3), (3,4)}: Ex. Graph the set of points {(1,3), (,3), (3,3)}: Horizontal Line Test Once you have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line crosses the graph more than once, then the graph does not define a one-to-one function. Ex 5. Select all of the following graphs which represent y as a function of x. a b c d e f

3 Function Notation The notation output = f(input) defines a function named f. This would be read output is f of input Ex1. The amount of garbage, G, produced by a city with population p is given by G f ( p). G is measured in tons per week, and p is measured in thousands of people. a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function f. b. Explain the meaning of the statement f 5. Ex4. Let ht () be the height above ground, in feet, of a rocket t seconds after launching. What is the input? What is the unit of the input? What is the output? Explain the meaning of each statement: a. h1 00 b. h 0 50 Ex0. Based on the table below, a. Evaluate f (8) b. Solve f ( x) 7 x f( x ) Ex 15. Given each function f( x ) graphed, evaluate f (0), f (1) and f (3) Ex 17. Given the function gx ( ) graphed here, a. Evaluate g () b. Solve gx

4 35. Suppose f x x 8x 4. Compute the following: a. f( 1) f(1) b. f ( 0) f ( ) 37. Let f t 3t 5 a. Evaluate f (0) b. Solve f t 0 Circles Q. What is the distance between the points P and Q? Definition of a Circle: A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point (h, k). The fixed distance r is called the radius, and The fixed point (h, k) is the center of the circle. Find the Equation of Circles r (h, k) (x, y) The equation of a circle centered at the point (h, k) with radius r can be written as ( x h) ( y k) r (Called the Standard form of a circle) Unit circle: If r = 1 and the center of the circle is at the origin, then the circle is called the unit circle. x + y = 1 Ex. Write the equation of the circle centered at (-9, 9) with radius 16. Write your answer in standard Form.

5 1. Domain and Range Domain and Range Domain: The set of possible input values to a function Range: The set of possible output values of a function Ex. Is {(4,-3), (5,4), (6,-3)} a function? Domain: Range: Ex. Is {(1,-1), (3,-3), (1,4)} a function? Domain: Range: Ex. Find the domain and range of the following Table? x 8 8 Domain: y Domain and Range of a graph Range: d a b Domain: c Range: Notation for Domain and Range Inequality Set Builder Notation Interval notation 5 h 10 h 5 h 10 (5, 10] 5 h 10 h 5 h 10 [5, 10) 5 h 10 h 5 h 10 (5, 10) h h 10 h h 10 [10, ) all real numbers h h (, ) Combining two intervals together: As an inequality it is: 1 x 3 or x 5 In set builder notation: x 1 x 3 or x 5 In interval notation: [1,3] (5, ) h h (,10)

6 Is the graph a function? Domain: Range: Is the graph a function? Domain: Range: Is the graph a function? Domain: Range: Is the graph a function? Domain: Range:

7 Is the graph a function? Domain: Range: Is the graph a function? Domain: Range: Finding the Domain of a Function The Domain of most frequently used Mathematical functions, like linear, quadratic, and exponential functions, is the set of all real numbers. However, the following situations are not: 1 A zero in the denominator, such as, and x A negative value under an even root symbol, such as x. a. f(x) = x + b. x g(x) = x + 1 x c. h(x) = x 1 d. i(x) = x 4 e. j(x) = 3x + 9 f. x 5 k(x) = 4 x x 5

8 Piecewise Defined Functions A piecewise defined function is a function that is defined by different formulas on different parts of its domain. Ex. Given each function, evaluate: f ( 1), f (0), f (), f (4). f x 3 4x if x 1 x 1 if x x if x 1 0 f x 4 if 0 x 3 3x 1 if x 3 Graph a piecewise defined function: (Find the Domain and Range of the functions) 4, if x. x + 4, if x < x, if x 1, if x Ex. f x f x x, if x f(x) = { x 1, if x 1 x, if < x 4 x, if x

9 1.3 Rates of Change and Behavior of Graphs Rate of Change A rate of change describes how the output quantity changes in relation to the input quantity. The units on a rate of change are output units per input units Some other examples of rates of change would be quantities like: A population of rats increases by 40 rats per week A barista earns $9 per hour (dollars per hour) Average Rate of Change The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values. Change of Output y y y1 Average rate of change = = Change of Input x x x Average Rate of Change using Function Notation Given a function f(x), the average rate of change on the interval [a, b] is Change of Output f ( b) f ( a) Average rate of change = Change of Input b a 1 Ex 1. Ex 1a. Ex 1b. Ex. Find the average rate of change of the following functions: a. f ( x) x x on the interval [1, 9] b. g( x) x 4 x on the interval [-1. 3]

10 on the interval [1, 1+h] d. x 3x x 1 c. i( x) 4 x f on the interval [ a, a h] e. x x 3x 4 g on the interval [x, x + h] f. j ( x) x 3 on the interval [ a, a h] 3 g. i x x x 5 on the interval [x, x + h] h. sx 3 5x on the interval [x, x + h]

11 Note: f is increasing on (a, b) and (c, d). f is decreasing on (b, c) Ex 3. Ex 4. a. b. Local Extrem A point where a function changes from increasing to decreasing is called a local maximum. A point where a function changes from decreasing to increasing is called a local minimum. Together, local maxima and minima are called the local extrema, or local extreme values, of the function.

12 Concavity A function is concave up if the rate of change is increasing. A function is concave down if the rate of change is decreasing. A point where a function changes from concave up to concave down or vice versa is called an inflection point. Larger increase Smaller increase Larger increase Smaller increase Graphically, concave down functions bend downwards like a frown, and concave up function bend upwards like a smile. Increasing Decreasing Concave Down Concave Up Ex 6. Examine the graph of the function and find the interval that is (a)concave up, (b)concave down and (c)the approximate coordinates, ( x, y), of all points of inflection (if any). Concave up Concave down Point of inflection Show Behaviors of the Toolkit Functions on P.43 and show how to use calculator to graph and estimate the local extrema (like the problem 40). Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down. / k( t) 3t t

13 1.4 Composition of Functions Composition of Functions When the output of one function is used as the input of another, we call the entire operation a composition of functions. We write f(g(x)), and read this as f of g of x or f composed with g at x. An alternate notation for composition uses the composition operator: ( f g)( x) is read f of g of x or f composed with g at x, just like f(g(x)). The domain of f g is the set of all x in the domain of g such that g(x) is in the domain of f. In other words, f gx is defined whenever both g(x) and f(g(x)) are defined. Ex1. Domain g(x) A f(g(x)) B f g x A B Ex5) If f x x and g x x, then find a) f(g(4)) e) gx f (Also find its domain) b) g(f()) c) f f 4 f) f x g (Also find its domain) d) f g1

14 For each pair of functions, find 1 x 4. f x, g x 4 x f g x and g f x and their domains. Simplify your answers. 7. If f(x) = x 4 + 6, g(x) = x 6 and h(x) = x, find f ( g( h( x ))). (Expand your answer) Find functions f( x ) and ( ) gx so the given function can be expressed as hx f g x 36. hx x h x 4 x.

15 1.5 Transformations of Functions In this section, we will learn to transform what we already know about the formulas or models, the tool-kit functions (P.43), called parent functions, into what we need. The transformations that we study are shifting, stretching, and reflecting. Ex1: Fill the tables and sketch the graphs of the equations. a) y x b) y x x y x y Domain: Range: g(x) = f(x) + h(x) = f(x) 3

16 1 Ex: Graph the equations y1 x, y 3x and y 3 x using the tables or calculator and draw a sketch of what you see in the coordinate 1 system below. Then compare the graphs of y 3x and y 3 x compare it with the graph of y1 x. (Domain of y is [,]) Domain: Range: g(x) = f(x) h(x) = f(x) + 3

17 Ex3: Fill the tables and sketch the graphs of the equations. Then compare them with the graph of y = x. a) y ( x ) b) y ( x 3) x y x y Domain: Range: g(x) = f(x + 3) h(x) = f(x 4) 3

18 Ex4: Graph the equations y1 x, y x and 3 y x using the tables or calculator and draw a sketch of what you see in the coordinate system below. Then compare the graphs of y x and y 3 x compare it with the graph of y1 x. Domain: Range: g(x) = f(x) g(x) = f ( x )

19 g(x) = f(x) h(x) = f( x) SUMMARY OF TRANSFORMATIONS: A, B, h, k determine the transformation of a function f(x): g( x) Af B( x h) k where any function f(x) has been transformed A: Vertical Stretch if A > 1 or Vertical Compression if 0 < A < 1 (Outside changes in y) -A: would constitute a Vertical Flip over the x axis B: Horizontal Stretch if 0<B<1 or Horizontal Compression if B> 1 (Inside changes in x) -B: would constitute a Horizontal Flip over the y axis h: Horizontal Shift (Inside changes in x) h>0 shifts left h<0 shifts right k: Vertical Shift (Outside changes in y) k>0 shifts up k<0 shifts down

20 40. For each equation below, determine if the function is Odd, Even, or Neither. a. f x x b. 4 3 g x x c. hx x x Ex5. Starting with the graph of f(x) = x, write the equation of the graph that results from a) shifting f(x) 7 units upward. y = b) shifting f(x) 6 units to the right. y = c) reflecting f(x) about the x-axis. y = Ex6. Given a function For example: f x, represent the following in function notation. f xshifted up 3 units: f x 3 a) f xshifted 3 units to the left: b) f xstretched horizontally by a factor of 4: 1 c) f xstretched vertically by a factor of : 3

21 Ex7. If the domain of f is [ 3,9] and range is [1,5], and g(x) = f(3x) 5, find the domain and range of g. Ex8. Use transformations to Graph f(x) = (x 3) + 1 (Use y = x, with domain [, ], to graph f.) Ex9. Use transformations to Graph f(x) = (x + ) + 3 (Use y = x, with domain [, ], to graph f.) The function f( x ) is graphed here. Write an equation for each graph below as a transformation of f( x ). f(x)

22 1.6 Inverse Functions Note: The idea of an inverse function is that the input becomes the output and the output becomes the input. One-to-one Functions A function is one-to-one if any two different inputs in the domain correspond to two different outputs in the range. That is, each X in the domain has one and only one Y in the range. Domain Range Horizontal Line Test Once you have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line crosses the graph more than once, then the graph does not define a one-to-one function. Ex1: Indicate if the function is a one-to-one function: a). b). c). Note: f reads f inverse or inverse of f, and f x f x If a function has an inverse, it is said to be invertible. Input and output relation: f(5) = f 1 (3) =

23 Ex. Ex3. a. b. Ex4. a. b. c.

24 Graph the inverse function Ex b. Ex a. Ex5: (a) Find f 1 x of 1 3x x 5 f x ; b) Find g 1 x of g x x 6 4 3x