Question

find the inverse function of f(x), f(x) = (x + 4)/(x - 3) f⁻¹(x) = (3x + 4)(x - 1) f⁻¹(x) = (3x + 4)/(x + 1) f⁻¹(x) = (3x - 4)/(x - 1) f⁻¹(x) = (3x + 4)/(x - 1)

Explanation:

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

We start with the function \( f(x)=\frac{x + 4}{x-3} \). Replace \( f(x) \) with \( y \), so we have \( y=\frac{x + 4}{x - 3} \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=\frac{y + 4}{y-3} \).

Step3: Solve for \( y \)

First, multiply both sides by \( y - 3 \) to get rid of the denominator: \( x(y - 3)=y + 4 \).
Expand the left - hand side: \( xy-3x=y + 4 \).
Move all terms with \( y \) to one side and the other terms to the opposite side: \( xy - y=3x + 4 \).
Factor out \( y \) from the left - hand side: \( y(x - 1)=3x + 4 \).
Then, divide both sides by \( x - 1 \) to solve for \( y \): \( y=\frac{3x + 4}{x - 1} \).
Since \( y = f^{-1}(x) \), we have \( f^{-1}(x)=\frac{3x + 4}{x - 1} \).

Answer:

\( f^{-1}(x)=\frac{3x + 4}{x - 1} \) (the fourth option: \( f^{-1}(x)=\frac{(3x + 4)}{(x - 1)} \))