For real numbers x where the functions are defined, let f(x) = \frac{x}{x^2 - 4} and g(x) … — ACT

For real numbers $x$ where the functions are defined, let $f(x) = \frac{x}{x^2 - 4}$ and $g(x) = \frac{x+3}{x-1}$ . What is the set of all real numbers $x$ for which the composite function $h(x) = f(g(x))$ is undefined?

A
$\left\{1\right\}$
B
$\left\{-2, 1, 2\right\}$
C
$\left\{1, 5\right\}$
$\left\{-\frac{1}{3}, 1, 5\right\}$ Answer
E
$\left\{-\frac{1}{3}, -2, 1, 2, 5\right\}$

Answer

The set containing negative one-third, one, and five

The correct answer is the set containing negative one-third, one, and five. The composite function

f(g(x))

is undefined when the inner function

g(x)

is undefined (which occurs at

x = 1

) or when the output of

g(x)

equals a value that makes the outer function

f(x)

undefined. The outer function

f(x)

is undefined when

x^2 - 4 = 0

, which gives

x = \pm 2

. Setting

g(x) = 2

leads to

x = 5

, and setting

g(x) = -2

leads to

x = -\frac{1}{3}

. Collecting all these values gives the complete set of restrictions.

Step-by-Step Solution

Determine the domain restriction of the inner function

x \neq 1

For the composite function

h(x) = f(g(x))

to be defined, the input

x

must first be in the domain of the inner function

g(x) = \frac{x+3}{x-1}

. Thus, the denominator cannot be zero, which gives

x - 1 \neq 0 \implies x \neq 1

Determine the restricted inputs for the outer function

u \neq 2

and

u \neq -2

The outer function

f(u) = \frac{u}{u^2 - 4}

is undefined when its denominator is zero. Solving

u^2 - 4 = 0

yields

u^2 = 4 \implies u = 2

u = -2

. Therefore, the output of the inner function,

g(x)

, cannot equal

2

-2

Solve the equation for the first restricted value of the outer function

x = 5

Setting the inner function equal to

2

gives

\frac{x+3}{x-1} = 2

. Multiplying both sides by

x-1

results in

x+3 = 2(x-1) \implies x+3 = 2x-2 \implies x = 5

Solve the equation for the second restricted value of the outer function

x = -\frac{1}{3}

Setting the inner function equal to

-2

gives

\frac{x+3}{x-1} = -2

. Multiplying both sides by

x-1

results in

x+3 = -2(x-1) \implies x+3 = -2x+2 \implies 3x = -1 \implies x = -\frac{1}{3}

Combine all restricted values of

x

S = \left\{-\frac{1}{3}, 1, 5\right\}

The values of

x

for which the composite function is undefined are

x = 1

(from the inner function's domain restriction) along with

x = 5

and

x = -\frac{1}{3}

(which make the inner function's output equal to the outer function's restricted inputs).

Key Concept

Domain of Composite Functions

Estimated Time:2m 30s

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