To verify that [tex]\(g(x) = \frac{1}{5}x + 5\)[/tex] is the inverse of [tex]\(f(x) = 5x - 25\)[/tex], we should check if [tex]\(f(g(x)) = x\)[/tex] and [tex]\(g(f(x)) = x\)[/tex].
To do so, let's go through the expressions given and determine if they reflect the necessary operations.
1. [tex]\(\frac{1}{5}\left(\frac{1}{5} x+5\right)+5\)[/tex]:
- Take [tex]\( x \)[/tex] and start with [tex]\( f(x) = 5x - 25 \)[/tex]
- The form of this expression does not directly apply to verifying [tex]\( g(x) \)[/tex] as the inverse of [tex]\( f(x) \)[/tex]
2. [tex]\(\frac{1}{5}(5 x-25)+5\)[/tex]:
- Take [tex]\( x \)[/tex] and apply the function [tex]\( f(x) \)[/tex] to it first:
[tex]\[ f(x) = 5x - 25 \][/tex]
- Then, apply the function [tex]\( g \)[/tex] to the result of [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = \frac{1}{5}(5x - 25) + 5 \][/tex]
- Simplify this expression step-by-step:
[tex]\[ \frac{1}{5}(5x - 25) + 5 = x - 5 + 5 = x \][/tex]
- This satisfies [tex]\( g(f(x)) = x \)[/tex], showing that [tex]\( g(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex].
3. [tex]\(\frac{1}{\left(\frac{1}{5} x+5\right)}\)[/tex]:
- This expression involves taking the reciprocal of [tex]\( g(x) \)[/tex], which is not directly related to proving inverse functions.
4. [tex]\(-\left(\frac{1}{5} x+5\right)+5\)[/tex]:
- This expression does not align with the operation of [tex]\( g(x) \)[/tex] on [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]\[ \frac{1}{5}(5x - 25) + 5 \][/tex]
