Question

1. what is the inverse function $f^{-1}(x)$ given $f(x) = 4x - 4$?

options:
$\bigcirc$ $f^{-1}(x) = \frac{x}{4} + 4$
$\bigcirc$ $f^{-1}(x) = \frac{x}{4} + 1$
$\bigcirc$ $f^{-1}(x) = \frac{x + 1}{4}$
$\bigcirc$ $f^{-1}(x) = 4x - 4$

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x) = 4x - 4 \). Replace \( f(x) \) with \( y \), so we have \( y = 4x - 4 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 4y - 4 \).

Step3: Solve for \( y \)

First, add 4 to both sides of the equation: \( x + 4 = 4y \). Then, divide both sides by 4: \( y=\frac{x + 4}{4}=\frac{x}{4}+1 \).

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

Now, replace \( y \) with \( f^{-1}(x) \) to get the inverse function. So, \( f^{-1}(x)=\frac{x}{4}+1 \).

Answer:

\( f^{-1}(x)=\frac{x}{4}+1 \) (corresponding to the option with this expression, likely the third option in the given choices as per the visible text \( f^{-1}(x)=\frac{x}{4}+1 \))