Let f(x) = x^3 + 1 and g(x) = sqrt(x). Find (fg)(x) and its domain.

To find \((fg)(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\) together.

Given:
- \(f(x) = x^3 + 1\)
- \(g(x) = \sqrt{x}\)

The product of these functions is:

\[
(fg)(x) = f(x) \cdot g(x) = (x^3 + 1) \cdot \sqrt{x}
\]

### Domain of \((fg)(x)\):

- \(f(x) = x^3 + 1\) is defined for all real numbers \(x\), so its domain is \(\mathbb{R}\).
- \(g(x) = \sqrt{x}\) is defined only when \(x \geq 0\) because the square root function is only real for non-negative values.

Since \((fg)(x)\) is defined only where both \(f(x)\) and \(g(x)\) are defined, the domain of \((fg)(x)\) will be:

\[
\text{Domain of } (fg)(x) = [0, \infty)
\]

Thus, the function is:

\[
(fg)(x) = (x^3 + 1) \cdot \sqrt{x}
\]

With the domain \([0, \infty)\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you find the derivative of \((fg)(x)\)?
2. What is the range of \((fg)(x)\)?
3. Can you simplify \((fg)(x)\) further?
4. How would the domain change if \(g(x) = \sqrt{1-x}\)?
5. Can you find \((fg)(x)\) for specific values of \(x\), like \(x = 4\) or \(x = 1\)?

**Tip:** When multiplying functions, the domain of the product is determined by the intersection of the domains of the individual functions.

Discover how to calculate (fg)(x) = (x^3 + 1) * sqrt(x) and determine its domain step-by-step. Understand the multiplication of functions and domain restrictions involved in the problem. Suitable for students in Grades 10-12.

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