To verify that [tex]\( g(x) = \frac{1}{5}x + 5 \)[/tex] is the inverse of [tex]\( f(x) = 5x - 25 \)[/tex], we need to check if the composition [tex]\( g(f(x)) \)[/tex] simplifies to [tex]\( x \)[/tex]. Let's examine this step-by-step:
1. Define the composition [tex]\( g(f(x)) \)[/tex]:
[tex]\[
g(f(x)) = g(5x - 25)
\][/tex]
Since [tex]\( g(x) = \frac{1}{5}x + 5 \)[/tex], we substitute [tex]\( 5x - 25 \)[/tex] in place of [tex]\( x \)[/tex]:
[tex]\[
g(5x - 25) = \frac{1}{5}(5x - 25) + 5
\][/tex]
2. Simplify the expression [tex]\( \frac{1}{5}(5x - 25) + 5 \)[/tex]:
[tex]\[
\frac{1}{5}(5x - 25) = \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 = x - 5
\][/tex]
Thus,
[tex]\[
\frac{1}{5}(5x - 25) + 5 = x - 5 + 5 = x
\][/tex]
So, the simplified expression [tex]\( \frac{1}{5}(5x - 25) + 5 \)[/tex] is indeed [tex]\( x \)[/tex]. This verifies that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Therefore, the given expression that verifies [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[
\frac{1}{5}(5x - 25) + 5
\][/tex]
