Let $f(x) = \sqrt { 7 - x } - 6$ and $f^{ –1}\left(x\right) = -\left(x + 6\right)^{2} + 7$.

1. State the domains and ranges of $f$ and $f^{ –1}$. If needed, restrict the domains so that they are inverses of each other.

   Answer (a):

   For $f$, the domain is $x ≤ −7$. The range is $y ≥ −6$. For the inverse, switch the domain and range.

2. Predict the output of $f^{ –1}\left(f\left(a\right)\right)$. Then check your prediction algebraically. Show all your work.

   Hint (b):

   Keep in mind the relationships between functions and their inverses.

   Answer (b):

   $f ^{–1}(f(a)) = a$