Chapter 10.4, Problem 39E

To determine

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

Answer to Problem 39E

Yes

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

Explanation of Solution

Given information: $f(x)=-\sqrt{3x+5}-2$

Formula Used: Horizontal line test, inverse relation and ${\left(a+b\right)}^2=a^2+2ab+b^2$

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.

  $f(x)=-\sqrt{3x+5}-2$

As,

  $y=f(x)$

Now,

  $y=-\sqrt{3x+5}-2$

So,

  $y+2=-\sqrt{3x+5}$

Now, squaring both sides

  ${\left(y+2\right)}^2={\left(-\sqrt{3x+5}\right)}^2$

So, simplify and use formula ${\left(a+b\right)}^2=a^2+2ab+b^2$

  $y^2+4y+4=3x+5$

Now,

  $x=\frac{y^2+4y+4-5}{3}$

So,

  $x=\frac{y^2+4y-1}{3}$

Hence,

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