To verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex], we need to show that [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex].
First, let's find [tex]\( f(g(x)) \)[/tex]:
1. Start with the function [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. Recall the function [tex]\( f(x) = 5x - 25 \)[/tex]:
[tex]\[
f\left(\frac{1}{5} x + 5\right) = 5\left(\frac{1}{5} x + 5\right) - 25
\][/tex]
4. Simplify the expression inside [tex]\( f(x) \)[/tex]:
[tex]\[
= 5 \cdot \frac{1}{5} x + 5 \cdot 5 - 25 = x + 25 - 25 = x
\][/tex]
This verifies that [tex]\( f(g(x)) = x \)[/tex].
Next, let's find [tex]\( g(f(x)) \)[/tex]:
1. Start with the function [tex]\( f(x) = 5x - 25 \)[/tex].
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(5x - 25)
\][/tex]
3. Recall the function [tex]\( g(x) = \frac{1}{5} x + 5 \)[/tex]:
[tex]\[
g(5x - 25) = \frac{1}{5}(5x - 25) + 5
\][/tex]
4. Simplify the expression inside [tex]\( g(x) \)[/tex]:
[tex]\[
= \frac{1}{5} \cdot 5x - \frac{1}{5} \cdot 25 + 5 = x - 5 + 5 = x
\][/tex]
This verifies that [tex]\( g(f(x)) = x \)[/tex].
Therefore, both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] are satisfied, confirming that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Among the provided expressions,
- [tex]\(\frac{1}{5}(5 x - 25) + 5\)[/tex] simplifies to [tex]\( x \)[/tex], which correctly represents [tex]\( g(f(x)) \)[/tex].
Therefore, the correct expression to verify that [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex] is:
[tex]\[
\frac{1}{5}(5 x - 25) + 5
\][/tex]
So, the answer is:
[tex]\[
\boxed{\frac{1}{5}(5 x - 25) + 5}
\][/tex]
