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What are the compositions of the given functions, [tex] f(x) [/tex] and [tex] g(x) [/tex], as directed in the problem?

Recall: When composing functions, substitute the expression of one function in for [tex] x [/tex] in the other function as directed by the given composition.

[tex]$
\begin{array}{l}
f(x) = 4x, \quad g(x) = x^2 - 1 \\
f \circ g(x) = ? \quad g \circ f(x) = ?
\end{array}
$[/tex]

A.
[tex]$
\begin{array}{l}
f \circ g(x) = 4x^2 - 4 \\
g \circ f(x) = 16x^2 - 1
\end{array}
$[/tex]

B.
[tex]$
\begin{array}{l}
f \circ g(x) = 4x^2 - 1 \\
g \circ f(x) = 4x^2 - 4
\end{array}
$[/tex]

C.
[tex]$
\begin{array}{l}
f \circ g(x) = 4x^2 + 1 \\
g \circ f(x) = 16x^2 - 1
\end{array}
$[/tex]

D.
[tex]$
\begin{array}{l}
f \circ g(x) = 4x^2 + 4 \\
g \circ f(x) = 4x^2 - 4
\end{array}
$[/tex]

Sagot :

GinnyAnswer
Let's break down the compositions of the functions step by step.

Given the functions:
[tex]\[ f(x) = 4x \][/tex]
[tex]\[ g(x) = x^2 - 1 \][/tex]

We need to find:
[tex]\[ f(g(x)) \][/tex]
[tex]\[ g(f(x)) \][/tex]

### Composition [tex]\( f(g(x)) \)[/tex]

1. Start with the inner function [tex]\( g(x) \)[/tex].
[tex]\[ g(x) = x^2 - 1 \][/tex]

2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex].
[tex]\[ f(g(x)) = f(x^2 - 1) \][/tex]

3. Use the definition of [tex]\( f \)[/tex]:
[tex]\[ f(x) = 4x \][/tex]
So,
[tex]\[ f(x^2 - 1) = 4(x^2 - 1) \][/tex]

4. Simplify the expression:
[tex]\[ 4(x^2 - 1) = 4x^2 - 4 \][/tex]

So,
[tex]\[ f(g(x)) = 4x^2 - 4 \][/tex]

### Composition [tex]\( g(f(x)) \)[/tex]

1. Start with the inner function [tex]\( f(x) \)[/tex].
[tex]\[ f(x) = 4x \][/tex]

2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex].
[tex]\[ g(f(x)) = g(4x) \][/tex]

3. Use the definition of [tex]\( g \)[/tex]:
[tex]\[ g(x) = x^2 - 1 \][/tex]
So,
[tex]\[ g(4x) = (4x)^2 - 1 \][/tex]

4. Simplify the expression:
[tex]\[ (4x)^2 - 1 = 16x^2 - 1 \][/tex]

So,
[tex]\[ g(f(x)) = 16x^2 - 1 \][/tex]

### Final Compositions

The compositions of the given functions are:
[tex]\[ f(g(x)) = 4x^2 - 4 \][/tex]
[tex]\[ g(f(x)) = 16x^2 - 1 \][/tex]

Thus, the correct answer is:

A. [tex]\( f \circ g(x) = 4x^2 - 4 \)[/tex]

[tex]\[ g \circ f(x) = 16x^2 - 1 \][/tex]
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