To find the inverse of the function [tex]\( f(x) = 5x \)[/tex], we need to follow these steps:
1. Definition of the Function:
We start with the given function [tex]\( f(x) = 5x \)[/tex].
2. Rewrite using [tex]\( y \)[/tex]:
To find the inverse function, we'll rewrite [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]:
[tex]\[
y = 5x
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
We need to solve this equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 5x
\][/tex]
Divide both sides of the equation by 5 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{y}{5}
\][/tex]
4. Express the Inverse Function:
Since [tex]\( x = \frac{y}{5} \)[/tex], we can write the inverse function [tex]\( f^{-1}(x) \)[/tex]. To match the notation, replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in our final expression:
[tex]\[
f^{-1}(x) = \frac{x}{5}
\][/tex]
5. Select the Correct Option:
Among the given options:
- [tex]\( f^{-1}(x) = -5x \)[/tex]
- [tex]\( f^{-1}(\pi) = -\frac{1}{5} x \)[/tex]
- [tex]\( f^{-1}(x) = \frac{1}{5} x \)[/tex]
- [tex]\( f^{-1}(x) = 5x \)[/tex]
The correct option is:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
Therefore, the correct answer is the third option:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
