Question

Find the function that corresponds to this inverse functionf(x)=3x+1/-5x-2, x ≠ -2/5

Answer 1

Answer::

$f^{-1}(x)=-\frac{1+2x}{3+5x},x\cancel{=}-\frac{3}{5}$

Explanation:

We want to find the function that corresponds to the inverse function:

$f(x)=\frac{3x+1}{-5x-2},x\cancel{=}-\frac{2}{5}$

To do this, we find the inverse of f(x).

In order to find the inverse of f(x), follow the steps below:

Step 1: Replace f(x) with y:

$y=\frac{3x+1}{-5x-2}$

Step 2: Swap x and y

$x=\frac{3y+1}{-5y-2}$

Step 3: Make y the subject of the equation:

$\begin{gathered} \text{ Cross multiply} \\ x(-5y-2)=3y+1 \\ \text{ Open the bracket on the left side:} \\ -5xy-2x=3y+1 \\ \text{ Bring all the terms containing y to one side of the equation.} \\ -5xy-3y=1+2x \\ \text{ Factor out y} \\ y(-5x-3)=1+2x \\ \text{ Divide both sides by }-5x-3 \\ y=\frac{1+2x}{-5x-3} \end{gathered}$

Step 4: Replace y the inverse of f(x).

$f^{-1}(x)=\frac{1+2x}{-5x-3}=\frac{1+2x}{-(5x+3)}=-\frac{1+2x}{3+5x}$

The function that corresponds to the given inverse function is:

$f^{-1}(x)=-\frac{1+2x}{3+5x},x\cancel{=}-\frac{3}{5}$