Chapter 5.1, Problem 122E

To determine

To find: inverse of the given function, if exist.

Answer to Problem 122E

The function has its inverse.

  $f^{-1}\left(x\right)=\pm \sqrt{x-3}+7$

Explanation of Solution

Given information :

A function is given as:

  $f\left(x\right)={\left(x-7\right)}^2+3$

Concept used:

A function has an inverse if the function is one to one and onto.

A vertical line should intersect the function once.

Inverse of any function can be found by considering $y=f\left(x\right)$ and interchanging $x$ by $y$ in thefunction, then expressing $y$ in terms of $x$ .

Calculation :

Consider the given function.

  $f\left(x\right)={\left(x-7\right)}^2+3$

Let’s apply vertical line test on the function as shown:

  

Here, the vertical line $x=10$ intersects the function once.

Therefore, the function $f\left(x\right)={\left(x-7\right)}^2+3$ has an inverse.

Now, evaluate the inverse as shown:

Consider $y=f\left(x\right)$

So,

  $y={\left(x-7\right)}^2+3$

Replace $x$ by $y$ .

  $x={\left(y-7\right)}^2+3$

Now, express $y$ in terms of $x$ .

  $\begin{array}{c}x-3={\left(y-7\right)}^2+3-3\\ {\left(y-7\right)}^2=x-3\\ y-7=\pm \sqrt{x-3}\\ y=\pm \sqrt{x-3}+7\end{array}$

Hence, $f^{-1}\left(x\right)=\pm \sqrt{x-3}+7$ .