Chapter 10.4, Problem 37E

To determine

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

Answer to Problem 37E

Yes

  $f^{-1}\left(x\right)=\frac{x^2}{9}+8$

Explanation of Solution

Given information: $f(x)=3\sqrt{x-8}$

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.

  $f(x)=3\sqrt{x-8}$

As,

  $y=f(x)$

So,

  $y=3\sqrt{x-8}$

Now, squaring both sides

  $y^2=9\times \left(x-8\right)$

So, simplify

  $x=\frac{y^2}{9}+8$

Hence,

  $f^{-1}\left(x\right)=\frac{x^2}{9}+8$