Which of the following functions has as its d… | GMAT Sprint

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PS · Problem Solving

705-805

Functions

Which of the following functions has as its domain the set of all real numbers?I. $f(x)=(64x)^\frac{1}{3}$ II. $f(x)=\sqrt{2x^2-8x+8}$ III. $f(x)=\frac{2}{2x^2-1}$

Answer Choices

Correct answer marked below

I only

II only

III only

I and II only

Correct

I, II, and III

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Explanation

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Official Explanation

Official

f(x)=(64x)^\frac{1}{3}

is equivalent to

f(x)=\sqrt[3]{64x} = 4\sqrt[3]{x}

. The cube root is defined for all real numbers, so its domain is all real numbers.

II.

f(x)=\sqrt{2x^2-8x+8}

. Simplify inside the square root:

2x^2-8x+8 = 2(x^2-4x+4) = 2(x-2)^2

. So

f(x)=\sqrt{2(x-2)^2}

. Since

(x-2)^2 \ge 0

for all real x, the expression under the square root is always non-negative. Thus the domain is all real numbers.…

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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.

Which of the following functions has as its domain the set of all real numbers?­­­­­­I. f(x)=(64x)13 f(x)=(64x)^\frac{1}{3}f(x)=(64x)31​II. f(x)=2x2−8x+8 f(x)=\sqrt{2x^2-8x+8}f(x)=2x2−8x+8​III. f(x)=22x2−1f(x)=\frac{2}{2x^2-1}f(x)=2x2−12​