Question

a composite function can be written as $w(u(x))$, where $u$ and $w$ are basic functions.

is $f(x) = \tan^4(x)$ a composite function? if so, what are $u$ and $w$

choose 1 answer:

a $f$ is composite. $u(x) = \tan(x)$ and $w(x) = x^4$.

b $f$ is composite. $u(x) = x^4$ and $w(x) = \tan(x)$.

c $f$ is not a composite function.

Explanation:

Step1: Recall composite function definition

A composite function \( w(u(x)) \) means we first apply \( u \) to \( x \), then apply \( w \) to the result.

Step2: Analyze \( f(x)=\tan^4(x) \)

Let's check option A: If \( u(x)=\tan(x) \), then \( u(x) \) gives \( \tan(x) \). Then \( w(u(x)) = w(\tan(x)) \). If \( w(x)=x^4 \), then \( w(\tan(x)) = (\tan(x))^4=\tan^4(x) \), which matches \( f(x) \).
Check option B: If \( u(x)=x^4 \), then \( w(u(x))=w(x^4)=\tan(x^4) \), which is not \( \tan^4(x) \). So B is wrong.
Option C is wrong because we just showed it is composite.

Answer:

A. \( f \) is composite. \( u(x) = \tan(x) \) and \( w(x) = x^4 \).