Use$f ( x ) = 3 + \sqrt { 2 x - 1 }$ to complete parts (a) through (e) below.

1. What are the domain and range of $f\left(x\right)$?

   Hint (a):

   Which values of x will cause the square root expression to be undefined?

2. What is the inverse of $f\left(x\right)$? Call it $g\left(x\right)$.

   Hint (b):

   If $f\left(x\right) = y$, then switch the $x$ and the $y$ in $f\left(x\right)$. Solve for $y$.

   Answer (b):

   $\textit{g}(\textit{x})=\frac{(\textit{x}-3)^{2}+1}{2}$

3. What are the domain and range of $g\left(x\right)$?

   Hint (c):

   Switch the domain and range of f(x).

4. Find an expression for f(g(x)).

   Hint (d):

   Substitute the equation g(x) that you found in part (b) for x in f(x).

   Step (d):

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

   Simplify.

5. Find an expression for $g\left(f\left(x\right)\right)$. What do you notice? Why does this happen?

   Hint (e):

   See part (d). What relationship do f(x) and g(x) have with each other? (What is the relationship between two inverses?)