The function f is defined by f(t) = 2^t for all real numbers t, and the binary operation \… — GRE General Test

The function $f$ is defined by $f(t) = 2^t$ for all real numbers $t$ , and the binary operation $\odot$ is defined for all non-zero real numbers $m$ and $n$ by $m \odot n = \frac{m}{n} - \frac{n}{m}$ . What is the value of $f(2) \odot f(-2)$ ?

$\frac{255}{16}$ Answer
B
$\frac{15}{16}$
C
$\frac{15}{4}$
D
$0$
E
$-\frac{255}{16}$

Answer

\frac{255}{16}

Evaluating

f(2)

yields

2^2 = 4

, and evaluating

f(-2)

yields

2^{-2} = \frac{1}{4}

. Substituting these values into the binary operation rule

m \odot n = \frac{m}{n} - \frac{n}{m}

gives

\frac{4}{1/4} - \frac{1/4}{4} = 16 - \frac{1}{16} = \frac{255}{16}

Step-by-Step Solution

Evaluate the function

f(t) = 2^t

t = 2

and

t = -2

f(2) = 2^2 = 4

and

f(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}

Negative exponents follow the rule

x^{-a} = \frac{1}{x^a}

Substitute

m = f(2) = 4

and

n = f(-2) = \frac{1}{4}

into the custom operation definition

m \odot n = \frac{m}{n} - \frac{n}{m}

4 \odot \frac{1}{4} = \frac{4}{\frac{1}{4}} - \frac{\frac{1}{4}}{4}

Apply the defined binary operation rule.

Simplify the complex fraction terms.

\frac{4}{\frac{1}{4}} = 4 \times 4 = 16

and

\frac{\frac{1}{4}}{4} = \frac{1}{4 \times 4} = \frac{1}{16}

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Subtract the two simplified terms.

16 - \frac{1}{16} = \frac{256}{16} - \frac{1}{16} = \frac{255}{16}

Find a common denominator to compute the final value.

Key Concept

Function evaluation with exponent rules combined with custom binary symbol operations

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