Answer
$f(g(x)) = x - 3$
$g(f(x)) = x - 6$
Work Step by Step
For these types of problems, we are asked to evaluate composite functions. We will use the inner function and substitute it where we see $x$ in the outer function.
To evaluate $f(g(x))$, we begin by plugging in the function $g(x)$ where we see $x$ in the outer function, $f(x)$:
$f(g(x)) = \frac{(2x - 3) - 3}{2}$
Combine like terms in the numerator:
$f(g(x)) = \frac{2x - 6}{2}$
Divide both numerator and denomination by greatest common factor, which is $2$, in this case:
$f(g(x)) = x - 3$
To evaluate $g(f(x))$, we begin by plugging in the function $f(x)$ where we see $x$ in the outer function, $g(x)$:
$g(f(x)) = 2(\frac{x - 3}{2}) - 3$
Multiply to simplify:
$g(f(x)) = \frac{2x - 6}{2} - 3$
Reduce the fraction by dividing numerator and denominator by $2$:
$g(f(x)) = (x - 3) - 3$
Combine like terms:
$g(f(x)) = x - 6$
