For all real numbers x and y, the custom binary operator \odot is defined by x \odot y = x… — GMAT

For all real numbers $x$ and $y$ , the custom binary operator $\odot$ is defined by $x \odot y = x^2 y - x y^2$ . The function $f$ is defined by $f(x) = x \odot 3$ . What is the value of $f(f(2))$ ?

Answer: 162

Answer

The value of

f(f(2))

is 162.

To evaluate the nested function

f(f(2))

, first determine

f(2)

using the definition

f(x) = x \odot 3

. Applying the definition of the custom operator

x \odot y = x^2 y - x y^2

with

x = 2

and

y = 3

yields

2 \odot 3 = (2)^2(3) - (2)(3)^2 = 12 - 18 = -6

. Then, evaluate

f(-6) = -6 \odot 3 = (-6)^2(3) - (-6)(3)^2 = 108 - (-54) = 162

Step-by-Step Solution

Evaluate the inner function expression f(2)

f(2) = 2 \odot 3 = (2)^2(3) - (2)(3)^2 = 12 - 18 = -6

By definition, f(x) = x \odot 3. Substituting x = 2 gives 2 \odot 3, which applies the custom operator definition x^2 y - x y^2.

Evaluate the outer function expression f(f(2)) = f(-6)

f(-6) = -6 \odot 3 = (-6)^2(3) - (-6)(3)^2 = 36(3) - (-54) = 108 + 54 = 162

Substitute the inner result -6 into the function f(x).

Key Concept

Custom Binary Operators and Nested Function Evaluation

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