Sure, let's find the composition of the functions [tex]\( g \circ f \)[/tex], which means [tex]\( g(f(x)) \)[/tex].
Given:
[tex]\[ f(x) = 2x + 1 \][/tex]
[tex]\[ g(x) = 3 - x \][/tex]
To find [tex]\( g(f(x)) \)[/tex], we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
1. First, compute [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 2x + 1 \][/tex]
2. Next, take this expression and substitute it into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(2x + 1) \][/tex]
3. Replace [tex]\( x \)[/tex] in the expression for [tex]\( g(x) \)[/tex] with [tex]\( 2x + 1 \)[/tex]:
[tex]\[ g(2x + 1) = 3 - (2x + 1) \][/tex]
4. Simplify the expression:
[tex]\[ 3 - (2x + 1) = 3 - 2x - 1 \][/tex]
[tex]\[ = 2 - 2x \][/tex]
So, [tex]\( g(f(x)) = 2 - 2x \)[/tex].
Therefore, the correct answer is:
[tex]\[ 2 - 2x \][/tex]
This matches the first option provided:
[tex]\[ -2x + 2 \][/tex]
The correct answer is:
[tex]\[ -2x + 2 \][/tex]
