Sec 2.1 Relations Learning Objectives: 1. Understand relations. 2. Find the domain and the range of a relation. 3. Graph a relation defined b an equation. 1. Understand relations Relation eists when the elements in one set are associated with element in a second set. If, are two elements in these sets and if a relation eists between and, then we sa that corresponds to or depends on and we write,. We can write as ordered pair ( ) Eample 1. Use the map to represent the relation as a set of ordered pairs. Then identif the domain and the range of the relation. Year NCAA Division I Women s Basketball Champion Ordered pair: 200 2004 2007 2006 Tennessee Connecticut Marland Balor Domain: Range: ------- 2. Find the domain and the range of a relation Definitions: 1. Domain of a relation is the set of all inputs ( s values) of the relation. 2. Range of a relation is the set of all outputs ( s values) of the relation. Eample 2. Find the domain and the range of the given relations 1. Domain: Range:
2. Set-builder notation Interval notation Domain: Range: ------- 3. Graph a relation defined b an equation Eample 3. Graph the relation = 2 + 2. Find the domain and the range. Label at least five points on the graph grid. (, ) 10-10 10 10 Set-builder notation Interval notation Domain: Range:
Sec 2.2 An Introduction to Functions Learning Objectives: 1. Determine whether a relation epressed as a map or ordered pairs represents a function. 2. Determine whether a relation epressed as an equation represents a function. 3. Determine whether a relation epressed as a graph represents a function. 4. Find the value of a function using function notation.. Work with applications of functions. 1. Determine Whether a Relation Epressed as a Map or Ordered Pairs Represents a Function Function is a relation in which each element in the domain (the value) of the relation corresponds to eactl one element in the range (the value) of the relation. Notation of a function f ( ) read f of or f evaluates at and f ( ) = where is called independent variable and is called dependent variable. Eample 1. Determine whether the relation represents a function. If the relation is a function, find the domain and range. 1. Number of square feet Selling Price 134 1976 2110 2698 $298, 70 $187,00 $209,000 $22,000 Domain: Range: 2, 2, 1,1, 0,2, 1,1, 0,3,,4 2. {( )( )( )( )( )( )} Answer: 2. Determine Whether a Relation Epressed as an Equation Represents a Function Eample 2. Determine whether the equation shows as a function of. 3 1. = + 1 2 Answer:
2. 3 = ± Answer: 3. Determine Whether a Relation Epressed as a Graph Represents a Function Vertical Line Test a set of points in the plane is the graph of a function if and onl if ever vertical line intersects the graph in at most one point. Eample 3. Determine whether the graph is that of a function. 1. 2. Answer: Answer: 4. Find the Value of a Function Eample 4. Let f( ) 1. f ( 2) = 2. Find the following. Answer: f +1 2. ( ) Answer: f + h 3. ( )
. Work with applications of functions Recall: f ( ) = where is called independent variable and is called dependent variable. Eample. The number N of cars produced at a certain factor in 1 da after t hours of operation is 2 given b ( t) 100t t = where 0 t 10 a. Identif the dependent variable and the independent variable. Dependent variable: Independent variable: b. Evaluate ( 4) and provide a verbal eplanation of the meaning of ( 4). Eplanation of the meaning of ( 4).
Sec 2.3 Functions and Their Graph Learning Objectives: 1. Graph functions in the librar of functions. 2. Find the domain of a function. 3. Obtain information from the graph of a function. 4. The zeros of a function. 1. Graph functions in the librar of functions 1. Linear Function: f ( ) = m+ b Domain: { R} = set of all real numbers; Range: { R} Slope = 0 m ; -intercept: ( 0, b) = set of all real numbers. 2 3 Eample: f ( ) = 3 2. Constant Function: f ( ) = b where b is an real numbers. Domain: { R} Range: { b} = set of all real numbers. Slope = = 0 m ; -intercept: ( 0, b) = = set consisting of a single number b. Graph is horizontal line. Eample: f ( ) = 3 (, ) 3. Identit Function: f ( ) = Domain: { R} Range: { R} = set of all real numbers; Slope = m = 1; = set of all real numbers. -intercept: ( 0, 0) Eample: f ( ) =
2 4. Square Function: f ( ) = = set of all real numbers; -intercept: ( 0, 0), - intercept: ( 0, 0) Domain: { R} Range: { 0} = set of nonnegative real numbers. 2 Eample: f ( ) = (, ) 3. Cube Function: f ( ) = Domain: { R} = set of all real numbers; -intercept: ( 0, 0), - intercept: ( 0, 0) Range: { R} = set of all real numbers. 3 Eample: f ( ) = (, ) 6. Square Root Function: f ( ) = 0 = set of nonnegative real numbers; Intercepts: ( 0, 0) Domain: { } Range: { 0} = set of nonnegative real numbers. Eample: f ( ) = (, )
7. Cube Root Function: f ( ) = 3 Domain: { R} Range: { R} = set of all real numbers; Intercepts: ( 0, 0) = set of all real numbers. Eample: f ( ) = 3 (, ) 1 = 0 = set of all nonzero real numbers; Intercepts: none 8. Reciprocal Function: f( ) Domain: { } Range: { 0} Eample: f( ) < 0 = set of all nonzero real numbers. 1 = > 0 (, ) (, ) 9. Absolute Value Function: f ( ) = Domain: { R} Range: { 0} = set of all real numbers; Intercepts: ( 0, 0) = set of nonnegative real numbers. Eample: f ( ) = (, )
2. Find the domain of a function Domain When the function f is given, then the domain ( s values) of the f is the largest set of real f ( s values) is a real number. numbers for which ( ) Eample 1. Find the domain of each function. Write answer using set-builder notation and interval notation. f 4 = 3 1. ( ) 2 Set-builder notation Interval notation ----------- 2. G ( ) = + 2 Set-builder notation Interval notation ----------- z+ 1 3. h ( z) = z 2 Set-builder notation Interval notation ----------- 3. Obtain information from the graph of a function Eample 2. Use the graph of the function g to answer the following. 1. Find g ( 4) ( 1, 4) ( 2, ) Answer: 2. Is g ( 1) positive or negative? ( 2, 0) ( 0, 2) ( 3, 0) 3. What is the domain of g? ( 4, 2) ( 4, ) 4. What is the range of g?
. For what numbers is g ( ) = 0 6. What are the -intercepts? 7. What is the -intercept? ---------- 4. The zeros of a function If f( c) = 0for some number c, then c is called a of a functionf. Eample 3. Determine if the value is a zero of the function. 1. f ( ) = 3 +12; 4 ----------- g = 2 2. ( ) + 2 ; 2 ----------- Eample 4. What are the zeros and intercepts of the function? Zeros: Intercepts:
Sec 2.4 Linear Functions and Models Learning Objectives: 1. Graph linear functions. 2. Find the zero of a linear function. 3. Build linear models from verbal descriptions 1. Graph Linear Functions Linear Function is a function of the form f ( ) m+ b graph of a linear function is a line with slope m and -intercept is (, b) Eample 1. Graph the linear equation f ( ) = 2 + 4 = where m and b are real numbers and the 0.. Label at least four points on the graph grid. 2. Find the zero of a linear function 1. If c is a zero of f, then f ( c) = 0 2. To find the zero of an function f, we solve the equation ( ) = 0 3. To find the zero of a linear function f ( ) m+ b Eample 2. Find the zero of f ( ) = 6 + 18 f. =, we solve the equation m +b= 0. 4 1 Eample 3. Suppose that f ( ) = + and g ( ) = + 1. 3 3 f = g. What is the value of f at the solution? a. Solve ( ) ( ) Value of f at the solution:
b. What point is on the graph of f? c. What point is on the graph of g? d. Solve ( ) g( ) f <. e. Graph f and g in the same Cartesian plane. Label the intersection point. Eample 4. Find a linear function f such that f ( 2 ) = 6 and ( ) = 12 f 2? f. What is ( ) Linear function: f ( 2) =
3. Build linear models from verbal descriptions (application of linear function) Eample. The cost, C, of renting a 12-foot moving truck for a da is $40 plus $0.3 times the number of miles driven. The linear function C ( ) = 0.3+ 40 describes the cost C or driving the truck miles. a. What is the implied domain of this linear function? Domain: b. Determine the C-intercept of the graph of the linear function. C-intercept: c. What is the rental cost if the truck is driven 80 miles? Write answer in a complete sentence. d. How man miles was the truck driven if the rental cost is $8.0? Write answer in a complete sentence. e. Graph the linear function. Label at least four points on the graph grid. C (, C)
Sec 2.7 Variation Learning Objectives: 1. Model and solve problems involving direct variation. 2. Model and solve problems involving inverse variation. 3. Model and solve problems involving combined or joint variation. 1. Model and Solve Problems Involving Direct Variation Suppose we let and represent two quantities. We sa that with, or is to, if there is a nonzero number k such that The number k is called the. Eample 1. Suppose that varies directl with. When = 2 then = 10. a. Find the constant of proportionalitk. b. Write the linear function relating the two variables in the form = k c. Find when = 4
Eample 2. Mortgage Pament The monthl pament p on a mortgage varies directl with the amount borrowed b. Suppose that ou decide to borrow $120,000 using a 1-ear mortgage at.% interest. You are told that our pament is $980.0. a. Write a linear function that relates the monthl pament p to the amount borrowed b for a mortgage with the same terms. b. Assume that ou have decided to bu a more epensive home that requires ou to borrow $10,000. What will our monthl pament be? Write answer in a complete sentence. c. Graph the relation between monthl pament and amount borrowed. Label at least four points on the graph grid. b p ( b, p) 2. Model and Solve Problems Involving Inverse Variation Suppose we let and represent two quantities. We sa that with, or is to, if there is a nonzero number k such that The number k is called the.
Eample 3. Suppose that varies inversel with. When = 20, then = 7. Find when = 10. a. Find the constant of proportionalit k. b. Write the function relating the variables. c. Find the indicated quantit. Eample 4. If the voltage in an electric circuit is held constant, the current l varies inversel with the resistance R. If the current is 60 amperes when the resistance is 10 ohms, find the current when the resistance is 120 ohms. 3. Model and Solve Problems Involving Combined or Joint Variation 1. When a variable quantit Q is proportional to the product of two or more other variables, we sa that Q with these quantities. Eample: = kz (read as varies jointl with and z ) 2. When direct and inverse variation occur at the same time, we have. k Eample: a. = (read as varies directl with and inversel with z ) z kmn b. = (read as varies jointl with m and n and inversel with z ) z
13 Eample. Suppose that Q varies directl with and inversel with. When Q =, = 9 and 12 = 4. Find Q when = 8 and = 7 a. Find the constant of proportionalit k. b. Write the function relating the variables. c. Find the indicated quantit. Eample 6. The Kinetic energ K of a moving object varies jointl with its mass and the square of the velocit v. If an object weighing 9 kilograms and moving with a velocit of 10 meters per second has a kinetic energ of 40 Joules. Find its kinetic energ with the velocit is 19 meters per second. Round the answer to the nearest Joule as needed.