For all real numbers x and y, the custom operation \Delta is defined by x \Delta y = (x + … — GMAT

For all real numbers $x$ and $y$ , the custom operation $\Delta$ is defined by $x \Delta y = (x + 1)(y - 2)$ . If the function $g$ is defined by $g(n) = n^2 - 3$ for all real numbers $n$ , what is the value of $g(3) \Delta g(2)$ ?

Answer: -7

Answer

The value of

g(3) \Delta g(2)

-7

Evaluating

g(3)

gives

3^2 - 3 = 6

, and evaluating

g(2)

gives

2^2 - 3 = 1

. Substituting these values as

x = 6

and

y = 1

into the custom operator definition

x \Delta y = (x + 1)(y - 2)

yields

(6 + 1)(1 - 2) = 7 \times (-1) = -7

Step-by-Step Solution

Evaluate the inner function

g(n)

n = 3

g(3) = 3^2 - 3 = 9 - 3 = 6

Before evaluating the custom operator, the inputs

g(3)

and

g(2)

must be explicitly calculated.

Evaluate the inner function

g(n)

n = 2

g(2) = 2^2 - 3 = 4 - 3 = 1

Determine the second operand for the custom operation.

Apply the custom operation

\Delta

to the results from Step 1 and Step 2

6 \Delta 1 = (6 + 1)(1 - 2) = 7 \times (-1) = -7

Substitute

x = 6

and

y = 1

into the formula

x \Delta y = (x + 1)(y - 2)

Key Concept

Evaluating functions and applying non-commutative custom operators

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