For all real numbers a and b, the custom operation \diamondsuit is defined by a \diamondsu… — GRE General Test

For all real numbers $a$ and $b$ , the custom operation $\diamondsuit$ is defined by $a \diamondsuit b = a^2 - b^2 + 2ab$ . The functions $f$ and $g$ are defined by $f(x) = x \diamondsuit 2$ and $g(x) = 2 \diamondsuit x$ . If $k$ is a positive real number such that $f(k) = g(k)$ , what is the value of $f(g(-1))$ ?

A
$-15$
$-7$ Answer
C
$-1$
D
$1$
E
$9$

Answer

-7

Applying the custom symbol rule gives

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

and

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

. Equating

f(k) = g(k)

yields

2k^2 = 8

, so the positive solution is

k = 2

. Evaluating the inner function gives

g(-1) = -(-1)^2 + 4(-1) + 4 = -1

. Substituting this value into

f

gives

f(-1) = (-1)^2 + 4(-1) - 4 = -7

Step-by-Step Solution

Express

f(x)

and

g(x)

using the definition of the custom operation

\diamondsuit

f(x) = x \diamondsuit 2 = x^2 - 2^2 + 2(x)(2) = x^2 + 4x - 4

and

g(x) = 2 \diamondsuit x = 2^2 - x^2 + 2(2)(x) = -x^2 + 4x + 4

Applying

a \diamondsuit b = a^2 - b^2 + 2ab

with

(a, b) = (x, 2)

and

(a, b) = (2, x)

separately.

Set

f(k) = g(k)

to solve for the positive constant

k

k^2 + 4k - 4 = -k^2 + 4k + 4 \implies 2k^2 = 8 \implies k^2 = 4 \implies k = 2

(since

k > 0

Equating the two algebraic function expressions and solving the resulting quadratic equation.

Evaluate the inner function expression

g(-1)

g(-1) = -(-1)^2 + 4(-1) + 4 = -1 - 4 + 4 = -1

Substituting

x = -1

into the formula for

g(x)

Evaluate the outer function

f(g(-1)) = f(-1)

f(-1) = (-1)^2 + 4(-1) - 4 = 1 - 4 - 4 = -7

Substituting the result from Step 3 into the formula for

f(x)

Key Concept

Evaluating algebraic custom operations, solving functional equalities, and applying nested function compositions.

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