To find the domain of the rational expression f(x) = \frac{x+3}{x^2 - 1} \div \frac{x^2 - … — ACT

To find the domain of the rational expression

f(x) = \frac{x+3}{x^2 - 1} \div \frac{x^2 - 9}{x^2 - 3x - 4}

a student must identify all values of

x

for which the expression is undefined. The expression is undefined for exactly five real values of

x

. What are these five values in increasing order?

Answer:From least to greatest, the values of

x

that make the expression undefined are:
1.

x =

【-3】
2.

x =

【-1】
3.

x =

【1】
4.

x =

【3】
5.

x =

【4】

Answer

The five real values of

x

that make the expression undefined, from least to greatest, are

-3

-1

1

3

, and

4

The rational expression is undefined when the denominator of the first expression is zero, when the denominator of the divisor is zero, or when the numerator of the divisor is zero. Factoring these three parts gives

(x^2 - 1) = (x-1)(x+1)

(x^2 - 3x - 4) = (x-4)(x+1)

, and

(x^2 - 9) = (x-3)(x+3)

. The values that make any of these equal to zero are

-3, -1, 1, 3, 4

Step-by-Step Solution

Identify values that make the first denominator zero.

x = 1

and

x = -1

The denominator of the first rational expression is

x^2 - 1

. Setting

x^2 - 1 = 0

yields

(x - 1)(x + 1) = 0

, which gives

x = 1

and

x = -1

Identify values that make the second denominator zero.

x = 4

and

x = -1

The denominator of the second rational expression (the divisor) is

x^2 - 3x - 4

. Setting

x^2 - 3x - 4 = 0

yields

(x - 4)(x + 1) = 0

, which gives

x = 4

and

x = -1

Identify values that make the divisor equal to zero.

x = 3

and

x = -3

Since the operation is division, dividing by zero is undefined. The divisor

\frac{x^2 - 9}{x^2 - 3x - 4}

is equal to zero when its numerator is zero. Setting

x^2 - 9 = 0

yields

(x - 3)(x + 3) = 0

, which gives

x = 3

and

x = -3

Combine all unique restricted values and sort them in ascending order.

-3, -1, 1, 3, 4

Combining the restricted values from steps 1, 2, and 3 gives the set

\{-3, -1, 1, 3, 4\}

. Sorting these from least to greatest yields

-3, -1, 1, 3, 4

Key Concept

Domain restrictions in division of rational expressions

Estimated Time:2m 30s

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