To find [tex]\( f(g(x)) \)[/tex], we need to perform the composition of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. The process can be detailed as follows:
1. We start with the functions:
[tex]\[
g(x) = x^2 - 3
\][/tex]
[tex]\[
f(x) = 3x - 4
\][/tex]
2. The composition [tex]\( f(g(x)) \)[/tex] means we want to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
3. First, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]. We have:
[tex]\[
f(g(x)) = f(x^2 - 3)
\][/tex]
4. Recall that [tex]\( f(x) = 3x - 4 \)[/tex]. To find [tex]\( f(x^2 - 3) \)[/tex], replace every occurrence of [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex] with [tex]\( x^2 - 3 \)[/tex]:
[tex]\[
f(x^2 - 3) = 3(x^2 - 3) - 4
\][/tex]
5. Next, distribute the 3 within the parentheses:
[tex]\[
3(x^2 - 3) = 3x^2 - 9
\][/tex]
6. Subtract 4 from the result:
[tex]\[
f(x^2 - 3) = 3x^2 - 9 - 4
\][/tex]
7. Combine like terms:
[tex]\[
3x^2 - 9 - 4 = 3x^2 - 13
\][/tex]
Therefore, the function [tex]\( f(g(x)) \)[/tex] is:
[tex]\[
f(g(x)) = 3x^2 - 13
\][/tex]
