Question

find the inverse function in slope - intercept form (mx + b):
$f(x)=\frac{3}{4}x - 6$
answer
$f^{-1}(x)=$

Explanation:

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

We start by writing the function as \( y = \frac{3}{4}x - 6 \). This is a standard step when finding the inverse of a function, as it makes it easier to swap the roles of \( x \) and \( y \).

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

Now we interchange \( x \) and \( y \) to get \( x = \frac{3}{4}y - 6 \). This is based on the definition of an inverse function, where if \( y = f(x) \), then \( x = f^{-1}(y) \), and we are working towards expressing \( y \) in terms of \( x \) for the inverse function.

Step3: Solve for \( y \) - Add 6 to both sides

We add 6 to both sides of the equation to isolate the term with \( y \). So, \( x + 6=\frac{3}{4}y \). This step is to get rid of the -6 on the right side, moving it to the left side with \( x \).

Step4: Solve for \( y \) - Multiply both sides by \( \frac{4}{3} \)

To solve for \( y \), we multiply both sides of the equation by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). So, \( y=\frac{4}{3}(x + 6) \). Expanding the right side, we get \( y=\frac{4}{3}x+8 \). This gives us the inverse function in slope - intercept form (\( mx + b \), where \( m=\frac{4}{3} \) and \( b = 8 \)).

Answer:

\( f^{-1}(x)=\frac{4}{3}x + 8 \)