For all non-zero real numbers a and b, the binary operator \star is defined by a \star b =… — GMAT

For all non-zero real numbers $a$ and $b$ , the binary operator $\star$ is defined by $a \star b = \frac{a^2 + b}{a}$ . The function $f$ is defined by $f(x) = x \star 16$ for all $x \neq 0$ . If $k$ is a positive real number such that $f(f(k)) = 10$ , what is the value of $k$ ?

Answer: 4

Answer

The value of

k

4

By definition of the custom operator,

f(x) = x + \frac{16}{x}

. Evaluating

f(4) = 4 + \frac{16}{4} = 8

and then

f(f(4)) = f(8) = 8 + \frac{16}{8} = 10

confirms that

k = 4

satisfies the condition.

Step-by-Step Solution

Express

f(x)

by applying the custom operator definition

f(x) = x \star 16 = \frac{x^2 + 16}{x} = x + \frac{16}{x}

Using

a = x

and

b = 16

a \star b = \frac{a^2 + b}{a}

simplifies the function formula.

Set

m = f(k)

and solve

f(m) = 10

for

m

m + \frac{16}{m} = 10 \implies m^2 - 10m + 16 = 0 \implies (m - 2)(m - 8) = 0

, yielding

m = 2

m = 8

Replacing the inner function evaluation with an intermediate variable creates a manageable quadratic equation.

Solve

f(k) = 2

and

f(k) = 8

for positive real values of

k

For

f(k) = 2

k^2 - 2k + 16 = 0

has discriminant

(-2)^2 - 4(1)(16) = -60 < 0

, giving no real solutions. For

f(k) = 8

k^2 - 8k + 16 = 0 \implies (k - 4)^2 = 0 \implies k = 4

Only

m = 8

produces a real value for

k

because

k + \frac{16}{k} \geq 8

for all positive real numbers

k

Key Concept

Custom Binary Operators and Nested Function Evaluation

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