Find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.
$f(x)=3 x-2$ and $g(x)=\frac{x+3}{2}$
a. $f(g(x))=$
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
b. $g(f(x))=$
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
c. $f$ and $g$ are inverses of each other.
$f$ and $g$ are not inverses of each other.

Step 1 ：Given the functions \(f(x) = 3x - 2\) and \(g(x) = \frac{x+3}{2}\), we need to find \(f(g(x))\) and \(g(f(x))\) and determine whether \(f\) and \(g\) are inverses of each other.

Step 2 ：To find \(f(g(x))\), we substitute \(g(x)\) into \(f(x)\). This gives us \(f(g(x)) = 3(\frac{x+3}{2}) - 2 = \frac{3x}{2} + \frac{5}{2}\).

Step 3 ：To find \(g(f(x))\), we substitute \(f(x)\) into \(g(x)\). This gives us \(g(f(x)) = \frac{3x - 2 + 3}{2} = \frac{3x}{2} + \frac{1}{2}\).

Step 4 ：If \(f\) and \(g\) were inverses of each other, we would have \(f(g(x)) = x\) and \(g(f(x)) = x\). However, neither \(f(g(x))\) nor \(g(f(x))\) are equal to \(x\).

Step 5 ：\(\boxed{\text{Final Answer:}}\)

Step 6 ：a. \(f(g(x)) = \frac{3x}{2} + \frac{5}{2}\)

Step 7 ：b. \(g(f(x)) = \frac{3x}{2} + \frac{1}{2}\)

Step 8 ：c. \(f\) and \(g\) are not inverses of each other.