For all non-zero real numbers a and b, the custom binary operation \diamondsuit is defined… — GRE General Test

For all non-zero real numbers $a$ and $b$ , the custom binary operation $\diamondsuit$ is defined by $a \diamondsuit b = \frac{a^2 - b^2}{ab}$ . If the function $f$ is defined for all $x \neq 0$ by $f(x) = x \diamondsuit 2$ , what is the value of $f(4) - f(1)$ ?

Answer: 3

Answer

Evaluating

f(4) = 4 \diamondsuit 2 = \frac{16 - 4}{8} = 1.5

and

f(1) = 1 \diamondsuit 2 = \frac{1 - 4}{2} = -1.5

, the required difference is

f(4) - f(1) = 1.5 - (-1.5) = 3

Step-by-Step Solution

Evaluate

f(4)

using the custom operation definition

f(4) = \frac{4^2 - 2^2}{4 \cdot 2} = \frac{12}{8} = 1.5

Substitute

a = 4

and

b = 2

into

a \diamondsuit b = \frac{a^2 - b^2}{ab}

Evaluate

f(1)

using the custom operation definition

f(1) = \frac{1^2 - 2^2}{1 \cdot 2} = -1.5

Substitute

a = 1

and

b = 2

into

a \diamondsuit b = \frac{a^2 - b^2}{ab}

Compute the difference

f(4) - f(1)

Subtracting

-1.5

from

1.5

yields

1.5 - (-1.5) = 1.5 + 1.5 = 3

Key Concept

Custom Symbol Operations and Function Evaluation

Estimated Time:1m 30s

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