The problem presents two functions, $f(x) = x^2$ and $g(x) = \sqrt{x} + 2$, and asks for the results of specific operations on these functions, along with their domains. Let's solve each part:

Part (a) $(f + g)(x)$

To find $(f + g)(x)$, we add the functions $f(x)$ and $g(x)$: $(f + g)(x) = f(x) + g(x) = x^2 + (\sqrt{x} + 2) = x^2 + \sqrt{x} + 2$

Domain of $(f + g)(x)$

The domain of $(f + g)(x)$ is restricted by $g(x)$, as the square root function $\sqrt{x}$ requires $x \geq 0$. Therefore, the domain of $(f + g)(x)$ is: $x \geq 0$

Part (b) $\left(\frac{f}{g}\right)(x)$

To find $\left(\frac{f}{g}\right)(x)$, we divide $f(x)$ by $g(x)$: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2}{\sqrt{x} + 2}$

Domain of $\left(\frac{f}{g}\right)(x)$

The domain of $\left(\frac{f}{g}\right)(x)$ is determined by the restrictions in both $f(x)$ and $g(x)$. Specifically:

1. $x \geq 0$, due to $\sqrt{x}$ in $g(x)$.
2. $\sqrt{x} + 2 \neq 0$, but since $\sqrt{x} + 2$ is always positive for $x \geq 0$, there are no further restrictions.

Thus, the domain of $\left(\frac{f}{g}\right)(x)$ is: $x \geq 0$

Summary of Results:

• (a) $(f + g)(x) = x^2 + \sqrt{x} + 2$ with domain $x \geq 0$
• (b) $\left(\frac{f}{g}\right)(x) = \frac{x^2}{\sqrt{x} + 2}$ with domain $x \geq 0$

Would you like a detailed explanation of each step, or do you have any further questions?

Here are some related questions:

1. What is the process for finding the domain of composite functions?
2. How can we determine the range of $f(x)$ and $g(x)$?
3. What happens to the domain if $g(x)$ has values that approach zero?
4. How does the square root function affect the domain of a function?
5. Can we simplify $\left(\frac{f}{g}\right)(x)$ further under specific values of $x$?

Tip: Always check each component of a function separately to determine the domain, especially when dealing with roots or fractions.