Let the function h be defined by h(x) = 3x - 5, and let the function g be defined by g(x) … — ACT

Let the function $h$ be defined by $h(x) = 3x - 5$ , and let the function $g$ be defined by $g(x) = x^2 + c$ for some constant $c$ . If $h(g(2)) = 10$ , what is the value of $c$ ?

Answer: 1

Answer

The value of the constant

c

1

The correct answer is

1

. Evaluating the inner function

g(2)

gives

2^2 + c = 4 + c

. Applying the outer function

h(x)

to this expression gives

h(4 + c) = 3(4 + c) - 5

. Simplifying this expression results in

12 + 3c - 5 = 7 + 3c

. We are given that this value equals

10

, so

7 + 3c = 10

. Subtracting 7 from both sides gives

3c = 3

, which yields

c = 1

Step-by-Step Solution

Evaluate

g(2)

in terms of

c

g(2) = 4 + c

We substitute

x = 2

into the definition

g(x) = x^2 + c

to get

2^2 + c = 4 + c

Substitute

g(2)

into the definition of

h(x)

to express

h(g(2))

h(g(2)) = 3(4 + c) - 5

Since the composition is

h(g(2))

, we evaluate the function

h

at the input value

g(2) = 4 + c

Set

h(g(2)) = 10

and solve the linear equation for

c

c = 1

Expanding and simplifying the equation

3(4 + c) - 5 = 10

gives

12 + 3c - 5 = 10

, which reduces to

7 + 3c = 10

. Subtracting 7 gives

3c = 3

, so

c = 1

Key Concept

Function Composition and Parameter Evaluation

Estimated Time:1m 15s

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