For all real numbers a and b, the custom binary operator \star is defined by a \star b = 2… — GMAT

For all real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = 2a - b$ . The function $f$ is defined for all real numbers by $f(x) = x^2 - 1$ . If $(x \star 3) \star f(3) = 6$ , what is the value of $x$ ?

A
-2
B
4
5Answer
D
6
E
3

Answer

First, evaluate the function

f(3) = 3^2 - 1 = 8

. Next, evaluate the first custom operation

x \star 3 = 2x - 3

. Substituting these into the overall expression yields

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 14

. Setting this equal to 6 gives

4x - 14 = 6

, which simplifies to

4x = 20

, so

x = 5

Step-by-Step Solution

Evaluate the inner function

f(3)

f(3) = 3^2 - 1 = 9 - 1 = 8

Substitute

x = 3

into the given definition

f(x) = x^2 - 1

Evaluate the inner custom operation

x \star 3

x \star 3 = 2x - 3

Apply the custom operator rule

a \star b = 2a - b

with

a = x

and

b = 3

Evaluate the outer custom operation

(x \star 3) \star f(3)

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 6 - 8 = 4x - 14

Substitute the results from Step 1 and Step 2 into the custom operator definition.

Solve the algebraic equation for

x

4x - 14 = 6 \implies 4x = 20 \implies x = 5

Set the simplified expression equal to 6 as given in the problem stem and isolate

x

Key Concept

Custom Operator Evaluation and Nested Functions

Estimated Time:1m 30s

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