To find the composite function [tex]\( g(f(x)) \)[/tex], let's proceed step-by-step:
1. Identify the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = 3x + 2
\][/tex]
[tex]\[
g(x) = x - 1
\][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] to find [tex]\( g(f(x)) \)[/tex]:
[tex]\[
g(f(x)) = g(3x + 2)
\][/tex]
3. Substitute [tex]\( 3x + 2 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(3x + 2) = (3x + 2) - 1
\][/tex]
4. Simplify the expression [tex]\( (3x + 2) - 1 \)[/tex]:
[tex]\[
g(3x + 2) = 3x + 2 - 1
\][/tex]
5. Combine like terms:
[tex]\[
g(3x + 2) = 3x + 1
\][/tex]
So, the composite function [tex]\( g(f(x)) \)[/tex] is:
[tex]\[
g(f(x)) = 3x + 1
\][/tex]
In the form [tex]\( ax + b \)[/tex], we have:
[tex]\[
g(f(x)) = 3x + 1
\][/tex]
Thus, [tex]\( a = 3 \)[/tex] and [tex]\( b = 1 \)[/tex].
