If f(x) = x + x^23x + 4, which of the followi… | GMAT Sprint

Free Practice Question

PS · Problem Solving

705-805

Functions

If $f(x) = x + \frac{x^2}{3x} + 4$ , which of the following numbers can't be in the domain of f(f(x))?
I. 3
II. 0
III. -3

Answer Choices

Correct answer marked below

I only

II only

III only

I and II

II and III

Correct

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Explanation

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Official Explanation

Official

First, simplify the function:

f(x) = x + \frac{x^2}{3x} + 4

.

Since

\frac{x^2}{3x} = \frac{x}{3}

for

x \neq 0

, we have

f(x) = x + \frac{x}{3} + 4 = \frac{4x}{3} + 4

, with the condition that

x \neq 0

.

Now consider

f(f(x))

. The domain of

f(f(x))

requires that:

x

is in the domain of

f

(so

x \neq 0

f(x)

is in the domain of

f

(so

f(x) \neq 0

Check each value:

I.

x = 3

f(3) = \frac{4 \cdot 3}{3} + 4 = 4 + 4 = 8 \neq 0

, and

3 \neq 0

. So

3

is in the domain.…

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What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.

If f(x)=x+x23x+4f(x) = x + \frac{x^2}{3x} + 4f(x)=x+3xx2​+4, which of the following numbers can't be in the domain of f(f(x))?I. 3II. 0III. -3