College Algebra 9th txtbk

178 CHAPTER 3 FUNCTIONS
Z Identifying Increasing and Decreasing Functions
We will now take a look at increasing and decreasing properties of functions. Informally,
a function is increasing over an interval if its graph rises as the x coordinate increases
(moves from left to right) over that interval. A function is decreasing over an interval if its
graph falls as the x coordinate increases over that interval. A function is constant on an
interval if its graph is horizontal (i.e., the height doesn’t change) over that interval (Fig. 5).
g(x)
5
f(x)
5
g(x) 2x 2
f(x) x 3
5
5
x
5
5
x
5
5
(a) Increasing on (, ) (b) Decreasing on (, )
h(x)
5
p(x)
5
h(x) 2
5
5
x
5
p(x) x 2 1
x
5
5
5
(c) Constant on (, )
(d) Decreasing on (, 0]
Increasing on [0, )
Z Figure 5 Increasing, decreasing, and constant functions.
More formally, we define increasing, decreasing, and constant functions as follows:
Z DEFINITION 1 Increasing, Decreasing, and Constant Functions
Let I be an interval in the domain of function f. Then,
1. f is increasing on I and the graph of f is rising on I if f(x 1 ) 6 f(x 2 )
whenever x 1 6 x 2 in I.
2. f is decreasing on I and the graph of f is falling on I if f(x 1 ) 7 f(x 2 )
whenever x 1 6 x 2 in I.
3. f is constant on I and the graph of f is horizontal on I if f(x 1 ) f(x 2 )
whenever x 1 6 x 2 in I.
Z Linear Functions
In Section 2-3, we studied the slope–intercept form of the equation of a line: y mx b,
where m is the slope, and b is the y intercept. We can carry over what we learned to the
study of linear functions.