Question

find the inverse function in slope-intercept form (mx+b):
$f(x) = \frac{5}{3}x + 15$

Explanation:

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

We start by writing the function as \( y = \frac{5}{3}x + 15 \). This is a standard step when finding the inverse of a function, as it helps us work with the equation in terms of \( x \) and \( y \) more easily.

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

To find the inverse, we interchange the roles of \( x \) and \( y \). So we get \( x = \frac{5}{3}y + 15 \). This is based on the definition of an inverse function, where if \( y = f(x) \), then \( x = f^{-1}(y) \), and we are essentially solving for the new \( y \) (which will be the inverse function) in terms of \( x \).

Step3: Solve for \( y \)

First, subtract 15 from both sides of the equation:
\( x - 15=\frac{5}{3}y \)
Then, multiply both sides by \( \frac{3}{5} \) to isolate \( y \):
\( y=\frac{3}{5}(x - 15) \)
Simplify the right - hand side:
\( y=\frac{3}{5}x-9 \)

Answer:

The inverse function of \( f(x)=\frac{5}{3}x + 15 \) in slope - intercept form is \( f^{-1}(x)=\frac{3}{5}x - 9 \)