Intermediate Algebra (11th edition)

(b)

Set the denominator equal to 0.

Factor.

or Zero-factor property

or Solve each equation.

The domain of gincludes all real numbers except 1 and 3, written

(c)

The denominator, 3, can never be 0, so the domain of hincludes all real numbers,

written is a real number.

(d)

Setting equal to 0 leads to There is no real number whose square

is Therefore, any real number can be used as a replacement for x. As in part (c),

the domain of ƒ is 5 x|xis a real number. 6

- 4.

x^2 + 4 x^2 =-4.

ƒ 1 x 2 =

2

x^2 + 4

5 x|x 6

h 1 x 2 =

8 x+ 2

3

5 x|xZ 1, 3 6.

x= 1 x= 3

x- 1 = 0 x- 3 = 0

1 x- 121 x- 32 = 0

x^2 - 4 x+ 3 = 0

g 1 x 2 =

3 +x

x^2 - 4 x+ 3

7.1 Rational Expressions and Functions; Multiplying and Dividing

NOW TRY

Values that make the

denominator 0 must

be excluded.

OBJECTIVE 3 Write rational expressions in lowest terms. In arithmetic, we

write the fraction in lowest terms by dividing the numerator and denominator by 5

to get We write rational expressions in lowest terms in a similar way, using the

fundamental property of rational numbers.

3

4.

15

20

Fundamental Property of Rational Numbers

If is a rational number and if cis any nonzero real number, then

That is, the numerator and denominator of a rational number may either be mul-

tiplied or divided by the same nonzero numberwithout changing the value of the

rational number.

a

b



ac

bc

.

a

b

Because is equivalent to 1, the fundamental property is based on the identity prop-

erty of multiplication.

NOTE A rational expression is a quotient of two polynomials. Since the value of a

polynomial is a real number for every value of the variable for which it is defined, any

statement that applies to rational numbers will also apply to rational expressions.

We use the following steps to write rational expressions in lowest terms.

c

c

NOW TRY
EXERCISE 1
For each rational function,
find all numbers that are not
in the domain. Then give the
domain, using set-builder
notation.

(a)

(b)

15

2 x^2 + 1

2 x- 1

x^2 - 4 x- 5

NOW TRY ANSWERS

1. (a)
   (b)none; is a real
   number 6

5 x|x

• 1, 5; 5 x|xZ-1, 5 6

Writing a Rational Expression in Lowest Terms

Step 1 Factorboth numerator and denominator to find their greatest com-

mon factor (GCF).

Step 2 Apply the fundamental property.Divide out common factors.