Chapter 7.2, Problem 26PPS

To determine

The inverse of the function $y={\left(x+1\right)}^2+3$ and graph the function and its inverse.

Answer to Problem 26PPS

The inverse of function $y={\left(x+1\right)}^2+3$ where $y=f\left(x\right)$ is $f^{-1}\left(x\right)=-1\pm \sqrt{x-3}$ .

Explanation of Solution

Given information: The given function is,

  $y={\left(x+1\right)}^2+\mathrm{3......}\left(\text{where }y=f\left(x\right)\right)$

Concept used: In order to find the inverse of a given function $f\left(x\right)$ write $f\left(x\right)=y$ and exchange the variables $\left(x\right)\text{ and }\left(y\right)$ and solve for the variable $\left(y\right)$ . At last write $y=f^{-1}\left(x\right)$ which is the required inverse of the function $f\left(x\right)$ .

Calculation: For the given function $y={\left(x+1\right)}^2+3$ , the inverse is evaluated as,

  $\begin{array}{c}y={\left(x+1\right)}^2+3\\ \text{or, }\left(x\right)={\left(\left(y\right)+1\right)}^2+\mathrm{3......}\left(\text{Exchanging variables}\right)\\ \text{or, }{\left(y+1\right)}^2=x-\mathrm{3......}\left(\text{Solving for variable }\left(y\right)\right)\\ y+1=\pm \sqrt{x-3}\end{array}$

Further simplifying,

  $\begin{array}{c}y=-1\pm \sqrt{x-3}\\ f^{-1}\left(x\right)=-1\pm \sqrt{x-3}\mathrm{......}\left(\text{Plugging }y=f^{-1}\left(x\right)\right)\end{array}$

Hence, the inverse of function $y={\left(x+1\right)}^2+3$ is $f^{-1}\left(x\right)=-1\pm \sqrt{x-3}$ .

Graph: The graph for the function $y={\left(x+1\right)}^2+3$ and its inverse function $f^{-1}\left(x\right)=-1\pm \sqrt{x-3}$ is shown as,