For all non-zero real numbers x and y, the binary operator \diamondsuit is defined by x \d… — GMAT

For all non-zero real numbers $x$ and $y$ , the binary operator $\diamondsuit$ is defined by $x \diamondsuit y = \frac{x}{y} - \frac{y}{x}$ . If the function $g$ is defined by $g(x) = x \diamondsuit 2$ for all non-zero real numbers $x$ , what is the value of $g(g(4))$ ?

$-\frac{7}{12}$ Answer
B
$\frac{7}{12}$
C
$\frac{3}{2}$
D
$-\frac{15}{4}$
E
$0$

Answer

-\frac{7}{12}

Evaluating the inner expression

g(4)

yields

4 \diamondsuit 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

. Evaluating

g\left(\frac{3}{2}\right)

yields

\frac{3/2}{2} - \frac{2}{3/2} = \frac{3}{4} - \frac{4}{3} = -\frac{7}{12}

, which is the correct final value.

Step-by-Step Solution

Evaluate the inner function expression

g(4)

using the given definition

g(x) = x \diamondsuit 2

g(4) = 4 \diamondsuit 2 = \frac{4}{2} - \frac{2}{4} = 2 - \frac{1}{2} = \frac{3}{2}

Nested function evaluations require evaluating the innermost expression first.

Substitute the result

g(4) = \frac{3}{2}

into the outer function to find

g\left(\frac{3}{2}\right)

g\left(\frac{3}{2}\right) = \frac{3}{2} \diamondsuit 2 = \frac{\frac{3}{2}}{2} - \frac{2}{\frac{3}{2}}

The output of the inner function becomes the input for the outer function.

Simplify the resulting fractions and subtract them.

\frac{3}{4} - \frac{4}{3} = \frac{9 - 16}{12} = -\frac{7}{12}

Finding a common denominator allows direct subtraction of fractions.

Key Concept

Nested Function Evaluation with Custom Operators

Estimated Time:1m 30s

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