Question

a composite function can be written as $w(u(x))$, where $u$ and $w$ are basic functions.
is $h(x) = \dfrac{\ln(x)}{\sqrt{x}}$ a composite function? if so, what are $u$ and $w?$
choose 1 answer:
a $h$ is composite. $u(x) = \ln(x)$ and $w(x) = \sqrt{x}$.
b $h$ is composite. $u(x) = \sqrt{x}$ and $w(x) = \ln x$.
c $h$ is not a composite function.

Explanation:

Step1: Recall Composite Function

A composite function is formed by applying one function to the result of another. But here \( h(x)=\frac{\ln(x)}{\sqrt{x}} \) is a quotient of two functions \( \ln(x) \) and \( \sqrt{x} \), not a composition (where one function is plugged into another, like \( w(u(x)) \) means \( w \) takes \( u(x) \) as input).

Step2: Analyze Options

• Option A: \( w(u(x)) \) would be \( \sqrt{\ln(x)} \), not \( \frac{\ln(x)}{\sqrt{x}} \).
• Option B: \( w(u(x)) \) would be \( \ln(\sqrt{x}) \), not \( \frac{\ln(x)}{\sqrt{x}} \).
• Option C: Since \( h(x) \) is a quotient of two functions, not a composition of one function into another, it's not a composite function.

Answer:

C. \( h \) is not a composite function.