For all real numbers x, the function g is defined by g(x) = cx + 5, where c is a constant.… — GRE General Test

For all real numbers $x$ , the function $g$ is defined by $g(x) = cx + 5$ , where $c$ is a constant. The custom operation $\diamond$ is defined for all real numbers $a$ and $b$ by $a \diamond b = g(a+b) - g(a-b)$ . If $3 \diamond 1 = 12$ , what is the value of $g(4)$ ?

Answer: 29

Answer

Applying the function definition

g(x) = cx + 5

to the custom operation yields

a \diamond b = [c(a+b)+5] - [c(a-b)+5] = 2bc

. Substituting

a=3

and

b=1

into

3 \diamond 1 = 12

gives

2(1)c = 12

, so

c = 6

. Consequently,

g(x) = 6x + 5

, and evaluating at

x = 4

yields

g(4) = 6(4) + 5 = 29

Step-by-Step Solution

Substitute

(a+b)

and

(a-b)

into the function definition

g(x) = cx + 5

to simplify

a \diamond b

a \diamond b = [c(a+b) + 5] - [c(a-b) + 5] = 2bc

Applying the definition of the custom binary operation in terms of function

g

eliminates the constant term

5

Use the given equality

3 \diamond 1 = 12

to determine the constant

c

2(1)c = 12 \implies 2c = 12 \implies c = 6

Plugging

a=3

and

b=1

into

2bc = 12

yields an equation in terms of

c

Evaluate

g(4)

using

c = 6

g(4) = 6(4) + 5 = 29

Substituting

x = 4

into

g(x) = 6x + 5

calculates the required numerical value.

Key Concept

Functions and Custom Symbol Operations

Estimated Time:1m 30s

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