Question

a composite function can be written as $w(u(x))$, where $u$ and $w$ are basic functions.
is $g(x) = (2x + 1)^4$ a composite function? if so, what are the \inner\ and \outer\ functions?
choose 1 answer:
a $g$ is composite. the \inner\ function is $x^4$ and the \outer\ function is $2x + 1$.
b $g$ is composite. the \inner\ function is $2x + 1$ and the \outer\ function is $x^4$.
c $g$ is not a composite function.

Explanation:

To determine if \( g(x) = (2x + 1)^4 \) is composite, we recall that a composite function \( w(u(x)) \) has an inner function \( u(x) \) (the input to the outer function) and an outer function \( w(x) \) (the function applied to the inner result). For \( g(x) \), if we let \( u(x) = 2x + 1 \) (inner, the expression inside the power) and \( w(x) = x^4 \) (outer, the power operation), then \( w(u(x)) = (2x + 1)^4 \), so it is composite. Option A swaps inner and outer, and option C is incorrect as it is composite.

Answer:

B. \( g \) is composite. The "inner" function is \( 2x + 1 \) and the "outer" function is \( x^4 \).