Answer
$f(g(x)) = \dfrac{x^2 + 5}{2}$
$g(f(x)) = \dfrac{x^2 + 10x + 25}{4}$
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, $f(x)$:
$f(g(x)) = \dfrac{x^2 + 5}{2}$
To evaluate $g(f(x))$, we begin by plugging in the function $f(x)$ where we see $x$ in the outer function, $g(x)$:
$g(f(x)) = \left(\dfrac{x + 5}{2}\right)^2$
Square the numerator and denominator:
$g(f(x)) = \dfrac{(x + 5)(x + 5)}{2^2}$
Use FOIL in the numerator to distribute the terms:
$g(f(x)) = \dfrac{x^2 + 5x + 5x + 25}{2^2}$
Evaluate exponent in the denominator:
$g(f(x)) = \dfrac{x^2 + 5x + 5x + 25}{4}$
Combine like terms:
$g(f(x)) = \dfrac{x^2 + 10x + 25}{4}$
