If [tex]\( f(x) = 5x \)[/tex], what is [tex]\( f^{-1}(x) \)[/tex]?

A. [tex]\( f^{-1}(x) = -5x \)[/tex]
B. [tex]\( f^{-1}(x) = -\frac{1}{5}x \)[/tex]
C. [tex]\( f^{-1}(x) = \frac{1}{5}x \)[/tex]
D. [tex]\( f^{-1}(x) = 5x \)[/tex]



Answer :

To determine the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 5x \)[/tex]:

1. Define the Function: We start with the given function:
[tex]\[ f(x) = 5x \][/tex]

2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]: For finding the inverse, we replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 5x \][/tex]

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]:
[tex]\[ y = 5x \][/tex]
[tex]\[ x = \frac{y}{5} \][/tex]

4. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the Inverse Function: We express the result of the inverse function with [tex]\( x \)[/tex] instead of [tex]\( y \)[/tex]. Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x}{5} \][/tex]

5. Simplify the Expression: To express it in a simplified form:
[tex]\[ f^{-1}(x) = \frac{1}{5}x \][/tex]

6. Identify the Correct Option: Comparing this result with the given options, the correct option is:
[tex]\[ f^{-1}(x) = \frac{1}{5}x \][/tex]

Therefore, [tex]\( f^{-1}(x) = \frac{1}{5}x \)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{3 \text{ (Corresponding to } f^{-1}(x) = \frac{1}{5}x \text{ )}} \][/tex]