Question

Find f^ -1(x) f(x)= (4x+3)^2+5

Answer 1

Given the following function:

$f\mleft(x\mright)=(4x+3)^2+5$

You can follow the steps shown below in order to find

$f^{-1}(x)$

Step 1. You need to rewrite the function using

$y=f(x)$

Then:

$y=(4x+3)^2+5$

Step 2. Solve for "x":

$\begin{gathered} y-5=(4x+3)^2 \\ \sqrt[]{y-5}=4x+3 \\ \sqrt[]{y-5}-3=4x \\ \\ x=\frac{\pm\sqrt[]{y-5}-3}{4} \\ \\ x=\frac{-3\pm\sqrt[]{y-5}}{4} \end{gathered}$

Step 3. Now you must exchange the variables:

$y=\frac{-3\pm\sqrt[]{x-5}}{4}$

Therefore, the answer is:

$f^{-1}(x)=\frac{-3\pm\sqrt[]{x-5}}{4}$