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.