Question

if ( f(x) = 5x ), what is ( f^{-1}(x) )?
options:
( f^{-1}(x) = -\frac{1}{5}x )
( f^{-1}(x) = 5x )
( f^{-1}(x) = -5x )
( f^{-1}(x) = \frac{1}{5}x )

Explanation:

Step1: Let \( y = f(x) = 5x \)

We start by setting \( y \) equal to the function \( f(x) \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \), so we get \( x = 5y \).

Step3: Solve for \( y \)

We solve the equation \( x = 5y \) for \( y \) by dividing both sides by 5. This gives \( y=\frac{1}{5}x \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

After solving for \( y \), we replace \( y \) with the inverse function notation \( f^{-1}(x) \), so \( f^{-1}(x)=\frac{1}{5}x \).

Answer:

\( f^{-1}(x)=\frac{1}{5}x \) (the option with \( f^{-1}(x)=\frac{1}{5}x \))