Question

which function is the inverse of $f(x) = -5x - 4$?
$f^{-1}(x) = 4x + 4$
$f^{-1}(x) = -\frac{1}{5}x + \frac{4}{5}$
$f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5}$
$f^{-1}(x) = -4x + 5$

Explanation:

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

We start with the function \( f(x) = -5x - 4 \). Replace \( f(x) \) with \( y \), so we have \( y = -5x - 4 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = -5y - 4 \).

Step3: Solve for \( y \)

First, we add 4 to both sides of the equation: \( x + 4 = -5y \). Then, we divide both sides by -5 to isolate \( y \): \( y = \frac{x + 4}{-5} \), which simplifies to \( y = -\frac{1}{5}x - \frac{4}{5} \).

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

Finally, we replace \( y \) with \( f^{-1}(x) \) to get the inverse function: \( f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5} \).

Answer:

\( f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5} \)