Question

Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.

Short answer

The steps required to find the inverse are:

1. interchange the variable $x$and $y$

2. solve for $y$

3. Substitute $f^{-1}\left(x\right)=y$.

4. Verify the obtained inverse.

Step-by-step solution

Step 1. Suppose an example 

Let an example be$f\left(x\right)=x^2$.

Step 2. Interchange the variable  xand y

Substituting $f\left(x\right)=y$, we get:

$y=x^2$

Interchanging the variable $x$and $y$, we get:

$x=y^2$

Step 3. solve for  y

Solving for $y$, we get:

role="math" localid="1644948675848" $$\begin{gathered} x=y^2 \\ y=\sqrt{x}[rootingbothsides] \end{gathered}$$

Substituting $y=f^{-1}\left(x\right)$, we get:

$f^{-1}\left(x\right)=\sqrt{x}$

Step 4. Verify that the functions are inverses.  

Substituting $f^{-1}\left(x\right)$in $f\left(f^{-1}\left(x\right)\right)=x$, we get:

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=x \\ f\left(\sqrt{x}\right)=x \\ {\left(\sqrt{x}\right)}^2=x \\ x=x \end{gathered}$$

This is true

Substituting $f\left(x\right)$in $f^{-1}\left(f\left(x\right)\right)=x$, we get:

role="math" localid="1644948900551" $$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=x \\ \sqrt{x^2}=x \\ x=x \end{gathered}$$

This is also true.

Thus, the obtained inverse is correct.