To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the function [tex]\( f(x) = 5x \)[/tex], we can use the following steps:
1. Express the function in terms of [tex]\( y \)[/tex]:
Let [tex]\( y = f(x) \)[/tex]. Therefore, we have:
[tex]\[
y = 5x
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse function, we interchange the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 5y
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Isolate [tex]\( y \)[/tex] by dividing both sides of the equation by 5:
[tex]\[
y = \frac{x}{5}
\][/tex]
4. Conclusion:
The inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{x}{5}
\][/tex]
To write it in a simplified form, we can also express it as:
[tex]\[
f^{-1}(x) = \frac{1}{5}x
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
f^{-1}(x) = \frac{1}{5} x
\][/tex]
