To verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to check that composing [tex]\( f \)[/tex] and [tex]\( g \)[/tex] yields the identity function [tex]\( x \)[/tex].
Given:
[tex]\[ f(x) = 5x - 25 \][/tex]
[tex]\[ g(x) = \frac{1}{5}x + 5 \][/tex]
To verify [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to compute [tex]\( f(g(x)) \)[/tex] and check if it simplifies to [tex]\( x \)[/tex].
Let's find [tex]\( f(g(x)) \)[/tex]:
1. Start by expressing [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{1}{5}x + 5 \][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{5}x + 5\right) \][/tex]
3. Use the expression for [tex]\( f \)[/tex]:
[tex]\[ f\left(\frac{1}{5}x + 5\right) = 5\left(\frac{1}{5}x + 5\right) - 25 \][/tex]
4. Simplify inside the parentheses:
[tex]\[ f\left(\frac{1}{5}x + 5\right) = 5 \cdot \frac{1}{5}x + 5 \cdot 5 - 25 \][/tex]
[tex]\[ f\left(\frac{1}{5}x + 5\right) = x + 25 - 25 \][/tex]
[tex]\[ f\left(\frac{1}{5}x + 5\right) = x \][/tex]
Since [tex]\( f(g(x)) = x \)[/tex], we have verified that [tex]\( g(x) \)[/tex] is indeed the inverse function of [tex]\( f(x) \)[/tex].
Therefore, the correct expression to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[ 5\left(\frac{1}{5} x+5\right)-25 \][/tex]
The corresponding multiple-choice option that matches this verification process is:
[tex]\[ 5\left(\frac{1}{5} x+5\right)+5 \][/tex]
Given that it was meant to match the expression logic in the multiple-choice options, there may have been a minor notation error in the choices provided, but based on the verification process used, it would indeed be:
[tex]\[ 5\left(\frac{1}{5} x+5\right)-25 \][/tex]
