1. Functions and Their Representations
Chapter 1.3
Functions and Their Representations

2. 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

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

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

5. 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}

6. 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

7. 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

8. 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

9. 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

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

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

12. 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

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

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

15. 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

16. 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

17. 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