To find the composition of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], which is denoted as [tex]\( (g \circ f)(x) \)[/tex], we need to evaluate [tex]\( g(f(x)) \)[/tex]. Here are the given functions:
[tex]\[
f(x) = 4x - 4
\][/tex]
[tex]\[
g(x) = 5x - 5
\][/tex]
Now, let's find the composition step-by-step:
1. First, find [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 4x - 4
\][/tex]
2. Next, substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
We want [tex]\( g(f(x)) \)[/tex], so we replace [tex]\( x \)[/tex] in the function [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex]. Thus,
[tex]\[
g(f(x)) = g(4x - 4)
\][/tex]
3. Evaluate [tex]\( g \)[/tex] at [tex]\( 4x - 4 \)[/tex]:
Recall the definition of [tex]\( g \)[/tex]:
[tex]\[
g(x) = 5x - 5
\][/tex]
Therefore, to find [tex]\( g(4x - 4) \)[/tex]:
[tex]\[
g(4x - 4) = 5(4x - 4) - 5
\][/tex]
4. Simplify the expression:
[tex]\[
g(4x - 4) = 5(4x - 4) - 5
\][/tex]
First, distribute the 5:
[tex]\[
g(4x - 4) = 20x - 20 - 5
\][/tex]
Combine like terms:
[tex]\[
g(4x - 4) = 20x - 25
\][/tex]
Therefore, the composition of the functions [tex]\( (g \circ f)(x) \)[/tex] is:
[tex]\[
(g \circ f)(x) = 20x - 25
\][/tex]
