College Algebra 9th txtbk

180 CHAPTER 3 FUNCTIONS
MATCHED PROBLEM 3
Find the slope and intercepts, and then sketch the graph of the linear function defined by
f (x) 3 2 x 6
Z Piecewise-Defined Functions
The absolute value function can be defined using the definition of absolute value from
Section 1-3:
x if x 6 0
f (x) x
x if x 0
Notice that this function is defined by different expressions for different parts of its domain.
Functions whose definitions involve more than one expression are called piecewise-defined
functions. Example 4 will show you how to work with a piecewise-defined function.
EXAMPLE 4 Analyzing a Piecewise-Defined Function
The function f is defined by
(A) Find f(3), f(2), f(1), and f(3).
f (x) 4x 11 if x 6 2
3 if 2 x 1
1 2 x 7 2 if x 7 1
(B) Graph f.
(C) Find the domain, range, and intervals where f is increasing, decreasing, or constant.
SOLUTIONS
(A) Since 3 is an x value less than 2, we use the formula 4x 11 to calculate f(3).
f(3) 4(3) 11 12 11 1
Since both 2 and 1 are in the interval 2 x 1, the output is 3 for both.
f(2) 3 and f(1) 3
Since 3 is an x value greater than 1, we use the formula 1 2 x 7 2 to calculate f (3).
f (3) 1 2 (3) 7 2 3 2 7 2 4 2 2
(B) To graph f, we graph each expression in the definition of f over the appropriate
interval. That is, we graph
y 4x 11
y 3
y 1 2 x 7 2
for x 6 2
for 2 x 1
for x 7 1
y
y 3
5
(2, 3) (1, 3)
5
1 7
y 2x 2
5
x
y 4x 11
5