Amanda wants to showcase her favorite function:$f ( x ) = 1 + \sqrt { x + 5 }$. She has built a function machine that performs these operations on the input values. Her brother Eric is always trying to mess up Amanda's stuff, so he created the inverse of $f(x)$, called it $e(x)$, and programmed it into a machine.

1. What is Eric's equation for his function $e(x)$?

   Answer (a):

   $e(x)=(x−1)^2−5$

2. What happens if the two machines are pushed together? What is $e(f(−4))$? Explain why this happens.

   Hint (b):

   Try several values for $x$ in $e(f(x))$. $x=−4$ is a very convenient value to use. So are $x=4$ and $x=−1$.
   What happens in each case?

3. If $f(x)$ and $e(x)$ are graphed on the same set of axes, what would be true about the two graphs?

   Hint (c):

   What characteristic has been common to all function-inverse pairs that you have graphed before?

4. Draw the two graphs on the same set of axes. Be sure to show clearly the restricted domain and range of Amanda's function.

   Hint (d):

   Since the domain of Amanda’s function is $x>−5$, the range of Eric’s inverse function will be $y>−5$.
   What is the range of Amanda’s function? How does this restrict the domain of Eric’s function which would otherwise look like a full parabola?