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

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

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

The sum of these functions is:

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

So:

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

### Domain of \((f+g)(x)\):

- \(f(x) = x^3 + 1\) 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 \((f+g)(x)\) is defined only where both \(f(x)\) and \(g(x)\) are defined, the domain of \((f+g)(x)\) will be:

\[
\text{Domain of } (f+g)(x) = [0, \infty)
\]

Thus, the function is:

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

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

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you find \((f-g)(x)\) and its domain?
2. What is the derivative of \((f+g)(x)\)?
3. Can you find the inverse of \((f+g)(x)\)?
4. How would the domain change if \(g(x) = \sqrt{x-1}\)?
5. What is the range of \((f+g)(x)\)?

**Tip:** When determining the domain of a function involving a square root, always check where the expression under the root is non-negative.

Learn how to find (f+g)(x) = x^3 + 1 + sqrt(x) and determine its domain. This problem involves function composition and domain analysis, covering key concepts in high school mathematics.

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