Sure! Let's work through the problem step-by-step to find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 5x \)[/tex].
1. Understand the Relationship:
The function [tex]\( f(x) = 5x \)[/tex] means that for each input [tex]\( x \)[/tex], the output is given by [tex]\( 5x \)[/tex].
2. Express [tex]\( f(x) \)[/tex] in Terms of [tex]\( y \)[/tex]:
We can rewrite the function as:
[tex]\[
y = 5x
\][/tex]
Here, [tex]\( y \)[/tex] is just another name for the output of the function.
3. Solve for [tex]\( x \)[/tex] in Terms of [tex]\( y \)[/tex]:
To find the inverse function, we need to solve this equation for [tex]\( x \)[/tex]. Start by isolating [tex]\( x \)[/tex]:
[tex]\[
y = 5x \implies x = \frac{y}{5}
\][/tex]
4. Express in Inverse Function Notation:
The expression [tex]\( x = \frac{y}{5} \)[/tex] tells us that for any output [tex]\( y \)[/tex] of the original function, the corresponding input [tex]\( x \)[/tex] is [tex]\( \frac{y}{5} \)[/tex]. Therefore, we can write the inverse function [tex]\( f^{-1}(x) \)[/tex] as:
[tex]\[
f^{-1}(x) = \frac{x}{5}
\][/tex]
5. Simplify the Expression:
Simplifying [tex]\( \frac{x}{5} \)[/tex], we can write it as:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
Among the given choices:
- [tex]\( f^{-1}(x) = -5x \)[/tex]
- [tex]\( f^{-1}(x) = -\frac{1}{5}x \)[/tex]
- [tex]\( f^{-1}(x) = \frac{1}{5}x \)[/tex]
- [tex]\( f^{-1}(x) = 5x \)[/tex]
The correct answer is [tex]\( f^{-1}(x) = \frac{1}{5} x \)[/tex].
Therefore, the correct answer is:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
