Decompose: 𝐹(𝑥) = √(2𝑥2 + 3)5


To decompose the function \( F(x) = \sqrt{(2x^2 + 3)^5} \), we can break it down into several nested functions, starting with the innermost part. Here’s how the decomposition can be done:

1. **Inner function**: \( g(x) = 2x^2 + 3 \)
   - This is the polynomial inside the composition.

2. **Exponentiation**: \( h(x) = g(x)^5 = (2x^2 + 3)^5 \)
   - This raises the result of \( g(x) \) to the 5th power.

3. **Square root**: \( f(x) = \sqrt{h(x)} = \sqrt{(2x^2 + 3)^5} \)
   - Finally, we take the square root of the function raised to the power of 5.

Thus, the function can be decomposed into the following steps:
\[
F(x) = \sqrt{(2x^2 + 3)^5} = f(h(g(x)))
\]

Would you like more details or have any questions about this decomposition?

Here are 5 related questions:
1. How would you differentiate \( F(x) \) using the chain rule?
2. What happens to \( F(x) \) as \( x \to \infty \)?
3. Can \( F(x) \) be simplified further?
4. How would you integrate \( F(x) \)?
5. What is the domain of \( F(x) \)?

**Tip**: Always check for the domain when dealing with square roots and exponents to ensure valid function evaluation.

Discover how to decompose the function F(x) = √(2x^2 + 3)^5 into its fundamental components. This step-by-step explanation covers algebraic concepts such as function decomposition, exponentiation, and square roots, along with related questions like differentiation and domain analysis. Suitable for students in Grades 10-12.

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