Answered

Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.