Determine whether the functions are inverses. a. $f(x)=\frac{x+6}{2}$ and $g(x)=2(x-6)$

Step 1 ：Given two functions, \(f(x)=\frac{x+6}{2}\) and \(g(x)=2(x-6)\)

Step 2 ：To determine if these functions are inverses of each other, we can compose the functions in both orders (f(g(x)) and g(f(x))) and see if the result is x in both cases.

Step 3 ：First, let's find f(g(x)). Substituting \(g(x)=2(x-6)\) into \(f(x)=\frac{x+6}{2}\), we get \(f(g(x))=\frac{2(x-6)+6}{2}=x-3\)

Step 4 ：Next, let's find g(f(x)). Substituting \(f(x)=\frac{x+6}{2}\) into \(g(x)=2(x-6)\), we get \(g(f(x))=2\left(\frac{x+6}{2}-6\right)=x-6\)

Step 5 ：The result of f(g(x)) is \(x - 3\) and the result of g(f(x)) is \(x - 6\). Since neither of these results is x, the functions f(x) and g(x) are not inverses of each other.

Step 6 ：\(\boxed{\text{The functions } f(x)=\frac{x+6}{2} \text{ and } g(x)=2(x-6) \text{ are not inverses of each other.}}\)