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

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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.

Express each of the following functions as the composite of two functions u and g so that f(x) = g[u(x)]: (a) f(x) = (x^2 + 5x + 1)^5, (b) f(x) = cos^3(x), (c) f(x) = sin(x^3).

Learn how to express functions like (x^2 + 5x + 1)^5, cos^3(x), and sin(x^3) as composite functions with step-by-step decomposition into inner and outer functions. This concept is key for advanced high school math students.

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