To verify if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to check if applying [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex] yields the original input, [tex]\( x \)[/tex]. In mathematical terms, we need to find out if [tex]\( g(f(x)) = x \)[/tex].
Given:
[tex]\[ f(x) = 5x - 25 \][/tex]
[tex]\[ g(x) = \frac{1}{5} x + 5 \][/tex]
Let's determine [tex]\( g(f(x)) \)[/tex]:
1. First, apply [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 5x - 25 \][/tex]
2. Now apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(5x - 25) \][/tex]
Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x - 25) = \frac{1}{5} (5x - 25) + 5 \][/tex]
Simplify the expression step-by-step:
[tex]\[ g(5x - 25) = \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 + 5 \][/tex]
[tex]\[ g(5x - 25) = x - 5 + 5 \][/tex]
[tex]\[ g(5x - 25) = x \][/tex]
We have shown that [tex]\( g(f(x)) = x \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex].
The correct expression to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
So, the correct answer is:
[tex]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
