Chapter 10, Problem 2CT

To determine

To find the inverse of the function.

Answer to Problem 2CT

Inverse of $f$ is $g\left(x\right)=\frac{{\left(x+1\right)}^2}{4}-3$

Explanation of Solution

Given information: $f(x)=2\sqrt{x+3}-1$

Formula Used: Inverse Relation

Calculation:

If a relation contains the point $\left(x,y\right)$ then the Inverse Relation contains the point $\left(y,x\right)$ because the inverse relation interchange the x and y coordinates.

  $f(x)=2\sqrt{x+3}-1$

As,

  $y=f(x)$

Now,

  $y=2\sqrt{x+3}-1$

So,

  $y+1=2\sqrt{x+3}$

Now,

  $\frac{y+1}{2}=\sqrt{x+3}$

So, squaring both sides

  ${\left(\frac{y+1}{2}\right)}^2={\left(\sqrt{x+3}\right)}^2$

Now,

  $x+3=\frac{{\left(y+1\right)}^2}{4}$

So,

  $x=\frac{{\left(y+1\right)}^2}{4}-3$

Hence,

Inverse of $f$ is $g\left(x\right)=\frac{{\left(x+1\right)}^2}{4}-3$