College Algebra 9th txtbk

170 CHAPTER 3 FUNCTIONS
In addition to evaluating functions at specific numbers, it is useful to be able to evaluate
functions at expressions that involve one or more variables. For example, the difference
quotient
f (x h) f (x)
h
x and x h in the domain of f, h 0
is very important in calculus courses.
EXAMPLE 6 Evaluating and Simplifying a Difference Quotient
For f(x) x 2 4x 5, find and simplify:
(A) f(x h) (B) f(x h) f(x) (C)
f (x h) f (x)
, h 0
h
SOLUTIONS
(A) To find f(x h), we replace x with x h everywhere it appears in the equation that
defines f and simplify:
f (x h) (x h) 2 4(x h) 5
x 2 2xh h 2 4x 4h 5
(B) Using the result of part A, we get
f (x h) f (x) x 2 2xh h 2 4x 4h 5 (x 2 4x 5)
x 2 2xh h 2 4x 4h 5 x 2 4x 5
2xh h 2 4h
(C)
f (x h) f (x)
h
2xh h2 4h
h
h(2x h 4)
h
Divide numerator and
denominator by h 0.
2x h 4
MATCHED PROBLEM 6 Repeat Example 6 for f(x) x 2 3x 7.
ZZZ CAUTION ZZZ
1. Remember, f(x h) is not a multiplication!
2. In general, f(x h) is not equal to f(x) f(h), nor is it equal to f(x) h.
Z Application
EXAMPLE 7 Construction
A rectangular feeding pen for cattle is to be made with 100 meters of fencing.
(A) If x represents the width of the pen, express its area A in terms of x.
(B) What is the domain of the function A (determined by the physical restrictions)?