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

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

   Hint (a):

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

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

   Hint (b):

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

   Answer (b):

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

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

   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 $g(f(x))$. What do you notice? Why does this happen?

   Hint (e):

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