1. Chapter 7
Functions and Graphs

2. 7.2 Domain and Range Determining the Domain and the Range
Restrictions on Domain
Piecewise-Defined Functions

3. Function
A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

4. Find the domain and range of the function f below.6 5 4 2 3 -4 1 -2 -1 -3 f -5
Here f can be written {(–5, 1), (1, 0), (3, –5), (4, 3)}.
The domain is the set of all first coordinates, {–5, 1, 3, 4}.
The range is the set of all second coordinates, {1, 0, –5, 3}.

5. For the function f represented below, determine each of the following.y x f 4 -2 -1 -4 -3 3 2 5 1 6 7
a) What member of the range is paired with -2
b) The domain of f
c) What member of the domain is paired with 6
d) The range of f

6. Solution a) What member of the range is paired with -2x y f 4 -2 -1 -4 -3 3 2 5 1 6
Input
Output 7
Locate -2 on the horizontal axis (this is where the domain is located). Next, find the point directly above -2 on the graph of f. From that point, look to the corresponding y-coordinate, 3. The “input” -2 has the “output” 3.

7. Solution b) The domain of fx y f 4 -2 -1 -4 -3 3 2 5 1 6
The domain of f 7
The domain of f is the set of all x-values that are used in the points on the curve. These extend continuously from −5 to 3 and can be viewed as the curve’s shadow, or projection, on the x-axis. Thus the domain is

8. Solution c) What member of the domain is paired with 6x y f 4 -2 -1 -4 -3 3 2 5 1 6
Input
Output 7
Locate 6 on the vertical axis (this is where the range is located). Next, find the point to the right of 6 on the graph of f. From that point, look to the corresponding x-coordinate, The “output” 6 has the “input”

9. Solution d) The range of fx y f 4 -2 -1 -4 -3 3 2 5 1 6
The range of f 7
The range of f is the set of all y-values that are used in the points on the curve. These extend continuously from -1 to 7 and can be viewed as the curve’s shadow, or projection, on the y-axis. Thus the range is

10. Determine the domain of
Solution
We ask, “Is there any number x for which we cannot compute 3x2 – 4?” Since the answer is no, the domain of f is the set of all real numbers.

11. Determine the domain of
Solution
We ask, “Is there any number x for which cannot be computed?” Since cannot
be computed when x – 8 = 0 the answer is yes.
To determine what x-value would cause x – 8 to be 0, we solve an equation:
x – 8 = 0,
x = 8
Thus 8 is not in the domain of f, whereas all other real numbers are. The domain of f is

12. Piecewise-Defined Functions
Some functions are defined by different equations for various parts of their
domains. Such functions are said to be piecewise-defined. For example, the function
given by f(x) = |x| can be described by
To evaluate a piecewise-defined function for an input a, we determine what part of the domain a belongs to and use the appropriate formula for that part of the domain.

13. Find each function value for the function f given by a. f(5) b
Find each function value for the function f given by a. f(5) b. f(–8) Solution a. Determine which equation to use. 5 is in the second part of the domain

14. Find each function value for the function f given by a. f(5) b
Find each function value for the function f given by a. f(5) b. f(–8) Solution b. Determine which equation to use. –8 is in the first part of the domain

15. Find each function value for the function f given by
a) f(3) b) f(2) c) f(7)
Solution
a) f(x) = x + 3: f(3) = 3 + 3 = 0
b)f(x) = x2; f(2) = 22 = 4
c)f(7) = 4x = 4(7) = 28