Chapter 2 Functions and Linear Functions

§ 2.1 Introduction to Functions

Blitzer, Intermediate Algebra, 4e – Slide #3 Basic Functions EXAMPLE Determine whether the following is a function. 1 drums guitar 2 violin 3 flute 4 Domain Range SOLUTION Yes, because none of the members of the domain correspond to more than one member of the range. Blitzer, Intermediate Algebra, 4e – Slide #3

Blitzer, Intermediate Algebra, 4e – Slide #4 Basic Functions EXAMPLE Determine whether the following is a function. 3 ants beetles 8 crickets 5 moths 9 Domain Range SOLUTION No, because one of the members of the domain, 9, corresponds to more than one member of the range. Blitzer, Intermediate Algebra, 4e – Slide #4

Blitzer, Intermediate Algebra, 4e – Slide #5 Basic Functions EXAMPLE Find the indicated function value: SOLUTION Replace x with 3 Evaluate the exponent Multiply Add and Subtract Blitzer, Intermediate Algebra, 4e – Slide #5

Blitzer, Intermediate Algebra, 4e – Slide #6 Basic Functions EXAMPLE Find the indicated function value: SOLUTION Replace z with -4 Evaluate the exponents Multiply Add Blitzer, Intermediate Algebra, 4e – Slide #6

Blitzer, Intermediate Algebra, 4e – Slide #7 Basic Functions EXAMPLE Find the indicated function value: SOLUTION Replace w with x+y Rewrite exponent Multiply Add Distribute NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!! Blitzer, Intermediate Algebra, 4e – Slide #7

Blitzer, Intermediate Algebra, 4e – Slide #8 Graphing Functions EXAMPLE Graph the function . SOLUTION x y = -3x+1 Ordered Pair (x,y) -2 f(x) = -3(-2) + 1 = 6 + 1 = 7 (-2,7) -1 f(x) = -3(-1) + 1 = 3 + 1 = 4 (-1,4) f(x) = -3(0) + 1 = 0 + 1 = 1 (0,1) 1 f(x) = -3(1) + 1 = -3 + 1 = -2 (1,-2) 2 f(x) = -3(2) + 1 = -6 + 1 = -5 (2,-5) Blitzer, Intermediate Algebra, 4e – Slide #8

Blitzer, Intermediate Algebra, 4e – Slide #9 Graphing Functions CONTINUED Blitzer, Intermediate Algebra, 4e – Slide #9

Blitzer, Intermediate Algebra, 4e – Slide #10 The Vertical Line Test EXAMPLE Use the vertical line test to identify graphs in which y is a function of x. (a) (b) (c) Blitzer, Intermediate Algebra, 4e – Slide #10

Blitzer, Intermediate Algebra, 4e – Slide #11 The Vertical Line Test CONTINUED SOLUTION (a) (b) (c) y is a function of x y is not a function of x y is a function of x Blitzer, Intermediate Algebra, 4e – Slide #11

Blitzer, Intermediate Algebra, 4e – Slide #12 Graphs of Functions EXAMPLE The figure shows the cost of mailing a first-class letter, f(x), as a function of its weight, x, in ounces. Use the graph to answer the following questions. Blitzer, Intermediate Algebra, 4e – Slide #12

Blitzer, Intermediate Algebra, 4e – Slide #13 Graphs of Functions CONTINUED (a) Find f (3). What does this mean in terms of the variables in this situation? (b) Find f (4). What does this mean in terms of the variables in this situation? (c) What is the cost of mailing a letter that weighs 1.5 ounces? (d) What is the cost of mailing a letter that weighs 1.8 ounces? Blitzer, Intermediate Algebra, 4e – Slide #13

Blitzer, Intermediate Algebra, 4e – Slide #14 Graphs of Functions CONTINUED SOLUTION (a) Find f (3). What does this mean in terms of the variables in this situation? f (3) = 0.83. This means that when a first-class letter weighs 3 ounces, postage costs 83 cents. Blitzer, Intermediate Algebra, 4e – Slide #14

Blitzer, Intermediate Algebra, 4e – Slide #15 Graphs of Functions CONTINUED (b) Find f (4). What does this mean in terms of the variables in this situation? f (4) = 1.06. This means that when a first-class letter weighs 4 ounces, postage costs $1.06. Blitzer, Intermediate Algebra, 4e – Slide #15

Blitzer, Intermediate Algebra, 4e – Slide #16 Graphs of Functions CONTINUED (c) What is the cost of mailing a letter that weighs 1.5 ounces? f (1.5) = 0.60. This means that when a first-class letter weighs 1.5 ounces, postage costs $0.60. Blitzer, Intermediate Algebra, 4e – Slide #16

Blitzer, Intermediate Algebra, 4e – Slide #17 Graphs of Functions CONTINUED (d) What is the cost of mailing a letter that weighs 1.8 ounces? f (1.8) = 0.60. This means that when a first-class letter weighs 1.8 ounces, postage costs $0.60. Blitzer, Intermediate Algebra, 4e – Slide #17

Obtaining Information from Graphs Graphs of Functions Obtaining Information from Graphs A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph. An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph. An arrow indicates that the graph extends indefinitely in the direction in which the arrow points. Blitzer, Intermediate Algebra, 4e – Slide #18

Blitzer, Intermediate Algebra, 4e – Slide #19 Domain and Range EXAMPLE Use the graph of the function to identify its domain and range. Blitzer, Intermediate Algebra, 4e – Slide #19

Blitzer, Intermediate Algebra, 4e – Slide #20 Domain and Range SOLUTION Domain = {x | x is a real number} Range = {y | y 3.6} Blitzer, Intermediate Algebra, 4e – Slide #20