Let the function f be given by f(x) = 3x - 5. If a second function g is defined in terms o… — SAT

Let the function $f$ be given by $f(x) = 3x - 5$ . If a second function $g$ is defined in terms of $f$ as $g(x) = 2f(x + 1) + 4$ , what is the value of $x$ for which $g(x) = 12$ ?

Answer: 2

Answer

Evaluating

g(x) = 12

gives

2f(x + 1) + 4 = 12

. Subtracting 4 from both sides yields

2f(x + 1) = 8

. Dividing by 2 yields

f(x + 1) = 4

. Substituting

x+1

into the definition

f(x) = 3x - 5

gives

3(x + 1) - 5 = 4

. Simplifying the equation results in

3x + 3 - 5 = 4

, which is

3x - 2 = 4

. Solving for

x

gives

3x = 6

, so

x = 2

Step-by-Step Solution

Set the equation

g(x) = 12

using the definition of

g(x)

2f(x + 1) + 4 = 12

We are given that

g(x) = 12

and want to find the corresponding value of

x

Isolate the function term

f(x + 1)

f(x + 1) = 4

Subtract 4 from both sides of the equation to get

2f(x + 1) = 8

, then divide by 2.

Use the definition of

f(x)

to express

f(x + 1)

f(x + 1) = 3(x + 1) - 5 = 3x - 2

Substitute

x + 1

in place of

x

in the function

f(x) = 3x - 5

Set the expression for

f(x + 1)

equal to 4 and solve for

x

x = 2

Solve the linear equation

3x - 2 = 4

by adding 2 to both sides to get

3x = 6

, then dividing by 3.

Key Concept

Applying multiple transformations to a linear function and solving the resulting equation using function notation.

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