To determine which expression could be used to verify if [tex]\( g(x) = \frac{1}{5} x + 5 \)[/tex] is the inverse of [tex]\( f(x) = 5x - 25 \)[/tex], we need to follow these steps:
1. Understand the Definitions:
- The function [tex]\( f(x) = 5x - 25 \)[/tex].
- The function [tex]\( g(x) = \frac{1}{5} x + 5 \)[/tex].
2. Verify Inverse Relationship:
- For [tex]\( g(x) \)[/tex] to be the inverse of [tex]\( f(x) \)[/tex], we must have:
[tex]\[
g(f(x)) = x \quad \text{and} \quad f(g(x)) = x
\][/tex]
3. Calculate [tex]\( g(f(x)) \)[/tex]:
[tex]\[
g(f(x)) = g(5x - 25)
\][/tex]
Substitute [tex]\( 5x - 25 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(5x - 25) = \frac{1}{5} (5x - 25) + 5 = x - 5 + 5 = x
\][/tex]
[tex]\( g(f(x)) = x \)[/tex], this part holds true.
4. Calculate [tex]\( f(g(x)) \)[/tex]:
[tex]\[
f(g(x)) = f\left( \frac{1}{5} x + 5 \right)
\][/tex]
Substitute [tex]\( \frac{1}{5} x + 5 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{1}{5} x + 5\right) = 5\left(\frac{1}{5} x + 5\right) - 25 = x + 25 - 25 = x
\][/tex]
[tex]\( f(g(x)) = x \)[/tex], this part also holds true.
5. List Possible Expressions:
Evaluate which of the given expressions simplifies to [tex]\( x \)[/tex]:
- [tex]\(\frac{1}{5}\left(\frac{1}{5} x + 5\right) + 5\)[/tex]
[tex]\[
\frac{1}{5}\left(\frac{1}{5} x + 5\right) + 5 = \frac{1}{25} x + 1 + 5 = \frac{1}{25} x + 6
\][/tex]
This does not simplify to [tex]\( x \)[/tex].
- [tex]\(\frac{1}{5}(5x - 25) + 5\)[/tex]
[tex]\[
\frac{1}{5}(5x - 25) + 5 = x - 5 + 5 = x
\][/tex]
This simplifies to [tex]\( x \)[/tex].
- [tex]\(\frac{1}{\left(\frac{1}{5} x + 5\right)}\)[/tex]
[tex]\[
\frac{1}{\left(\frac{1}{5} x + 5\right)} = \text{This simplifies to} \frac{1}{5}x + 5 \text{ if this is the inverse}
\][/tex]
This does not simplify to [tex]\( x \)[/tex].
- [tex]\( 5\left(\frac{1}{5} x + 5\right) + 5 \)[/tex]
[tex]\[
5\left(\frac{1}{5} x + 5\right) + 5 = x + 25 + 5 = x + 30
\][/tex]
This does not simplify to [tex]\( x \)[/tex].
Based on this analysis, the second expression, [tex]\(\frac{1}{5}(5x - 25) + 5 \)[/tex], is the one that can be used to verify if [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Therefore, the correct option is:
[tex]\[
\boxed{\frac{1}{5}(5x - 25) + 5}
\][/tex]
