To express each of the functions in the form $f(x) = g(u(x))$, we need to identify two functions $u(x)$ and $g(x)$ such that:

• $u(x)$ is the inner function,
• $g(u(x))$ is the outer function.

Let's decompose each part:

(a) $f(x) = \left(x^2 + 5x + 1\right)^5$

Here, the inner function can be the expression inside the parentheses, and the outer function can be raising something to the power of 5.

• $u(x) = x^2 + 5x + 1$
• $g(u) = u^5$

Thus, $f(x) = g(u(x)) = (u(x))^5 = \left(x^2 + 5x + 1\right)^5$.

(b) $f(x) = \cos^3(x)$

This can be written as the cube of the cosine function. The inner function is $\cos(x)$, and the outer function is cubing.

• $u(x) = \cos(x)$
• $g(u) = u^3$

Thus, $f(x) = g(u(x)) = (\cos(x))^3$.

(c) $f(x) = \sin(x^3)$

Here, the inner function is $x^3$, and the outer function is $\sin(x)$.

• $u(x) = x^3$
• $g(u) = \sin(u)$

Thus, $f(x) = g(u(x)) = \sin(x^3)$.

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How do you determine the inner and outer functions in a composite function?
2. Can every function be expressed as a composition of two functions?
3. How would you decompose $f(x) = (2x - 3)^7$ into a composite function?
4. What would the inner and outer functions be for $f(x) = \tan^4(x^2)$?
5. Can you express $f(x) = e^{x^2 + 4x}$ as a composite function?

Tip: Always identify the most nested operation to choose the inner function for easier decomposition.