Functions and Their Representations Chapter 1.3 Functions and Their Representations

Learn function notation Represent a function four different ways Define a function formally Identify the domain and range of a function Use calculators to represent functions (optional) Identify functions Represent functions with diagrams and equations

Teaching Example 1 Graph by hand. Solution Create a table of values.

Teaching Example 1 (continued) Now plot the points and join with a smooth curve. 4

Teaching Example 2 Solution Let f be the function defined by f(1) = 2, f(2) = 3, and f(3) = 4. Write f as ordered pairs. Give the domain and range. Solution f = {(1, 2), (2, 3), (3, 4)} D = {1, 2, 3}, R = {2, 3, 4}

The domain of f is all real numbers. Teaching Example 3 Let Find f(3), f(a), and f(a + 2). What is the domain of f ? Solution The domain of f is all real numbers. 6

Teaching Example 3 Solution The function f computes the U.S. mobile display and revenue for 2011 in millions of dollars: f(Apple) = 95, f(Google) = 150m=, and f(Yahoo) = 50. a. Write f as a set of ordered pairs. b. Give the domain and range of f. Solution a. f{(Apple, 95), (Google, 150), (Yahoo, 50)} b. D = {Apple, Google, Yahoo} R = {50, 95, 150} 7

Teaching Example 4 Solution Let a. If possible, evaluate f(2), f(1), and f(a + 1). b. Find the domain of f. Solution a. b. 8

Teaching Example 5 The graph of is shown below. (a) Find the domain and range of f. (b) Use the formula to evaluate f(−1). (c) Use the graph to evaluate f(−1). 9

Teaching Example 5 (cont) Solution (a) The domain of f is all real numbers. (b) (c) 10

Teaching Example 6 The graph of is shown here. Find the domain and range. Range Domain 11

Teaching Example 7 Solution When the relative humidity is 100%, air cools 1.1°F for every 1000-foot increase in altitude. Give verbal, symbolic, graphical, and numerical representations of a function ƒ that computes this change in temperature for an increase in altitude of x thousand feet. Let the domain of ƒ be 0 ≤ x ≤ 5. Solution Verbal: Multiply the input x by −1.1 to obtain y, the temperature change. 12

Teaching Example 7 (continued) Symbolic: Let f(x) = −1.1x. Graphical: [0, 5, 1] by [−8, 2, 1] 13

Teaching Example 7 (continued) Numerical: On the Table Setup screen, set Indpnt to Ask, since the domain is restricted to [0, 5]. 14

Teaching Example 8 Solution (a) Is f = {(1, 2), (2, 2), (3, 2)} a function? (b) Is g = {(2, 1), (2, 2), (2, 3)} a function? Solution (a) Yes (b) No, because the input value 2 has two output values. 15

Teaching Example 9 Solution Suppose the graph of f is a nonvertical line. Is f a function? Solution Since every vertical line that could be visualized would intersect the graph at most once, the graph represents a function. 16

Teaching Example 10 Solution Determine whether y is a function of x. (a) 4y = x (b) Solution (a) , which is a nonvertical line. Thus, y is a function of x. (b) represents a circle, which is not a function. 17