College Algebra 9th txtbk

SECTION 4–4 Rational Functions and Inequalities 299
5
y
Z DEFINITION 1 Rational Function
A function f is a rational function if it can be written in the form
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(1, 2)
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x
where p(x) and q(x) are polynomials.
f (x) p(x)
q(x)
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(a) f(x) (x 1)(x2 3)
x 1
y
When working with rational functions, we will assume that the coefficients of
p(x) and q(x) are real numbers, and that the domain of f is the set of all real
numbers x such that q(x) 0.
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If a real number c is a zero of both p(x) and q(x), then, by the factor theorem, x c is a
factor of both p(x) and q(x). The graphs of
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x
f (x) p(x)
q(x) (x c)p r(x)
(x c)q r (x)
and
f r (x) p r(x)
q r (x)
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(b) f(x) x 2 3
Z Figure 1
are then identical, except possibly for a “hole” at x c (Fig. 1).
Later in this section we will explain how to handle the minor complication caused by
common real zeros of p(x) and q(x). But to avoid that complication now,
unless stated to the contrary, we will assume that for any rational function f
we consider, p(x) and q(x) have no real zero in common.
Because a polynomial q(x) of degree n has at most n real zeros, there are at most n
real numbers that are not in the domain of f (x) P(x)q(x) . Because a fraction equals 0
only if its numerator is 0, the x intercepts of the graph of f are the real zeros of a polynomial
p(x) of degree m. So the number of x intercepts of f is at most m.
EXAMPLE 1 Domain and x Intercepts
Find the domain and x intercepts for f (x) 2x2 2x 4
x 2 .
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SOLUTION
f (x) p(x)
q(x) 2x2 2x 4 2(x 2)(x 1)
x 2
9 (x 3)(x 3)
Because q(x) 0 for x 3 and x 3, the domain of f is
x 3
or
(, 3) (3, 3) (3, )
Because p(x) 0 for x 2 and x 1, the zeros of f, and the x intercepts of f, are 1
and 2.
MATCHED PROBLEM 1
Find the domain and x intercepts for
f (x) 3x2 12
x 2 2x 3 .
The graph of the rational function
is shown in Figure 2 on the next page.
f (x) x2 1.44
x 3 x