Chapter 10.4, Problem 33E

To determine

To check horizontal line test whether inverse of $f$ is a function and find inverse relation.

Answer to Problem 33E

Yes

  $f^{-1}\left(x\right)=x^2-3$

Explanation of Solution

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

Formula Used: Horizontal line test and inverse relation

Calculation:

If the horizontal line intersects the graph of a function in all places at exactly one point, then the given function should have an inverse that is also a function. We say this function passes the horizontal line test.

  

As the horizontal line intersect with the graph of function at 1 point. So the function passes the horizontal test. So, the inverse of $f$ is a function.

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.

  $f(x)=\sqrt{x+3}$

As,

  $y=f(x)$

So,

  $y=\sqrt{x+3}$

Now, squaring both sides

  $y^2=x+3$

So, simplify

  $x=y^2-3$

Hence,

  $f^{-1}\left(x\right)=x^2-3$