The problem asks to find four operations between functions $f(x)$ and $g(x)$ for each pair provided:

• $(f + g)(x) = f(x) + g(x)$
• $(f - g)(x) = f(x) - g(x)$
• $(f \cdot g)(x) = f(x) \cdot g(x)$
• $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Additionally, you need to state the domain of each new function. The domain consists of all the $x$-values for which the result is defined. Division by zero and square roots of negative numbers must be avoided in the domain.

Let’s walk through one example (the first one) and guide you in solving the rest.

Example 1:

$f(x) = x^2 + 4$, $g(x) = \sqrt{x}$.

1. Addition:
   $(f + g)(x) = (x^2 + 4) + \sqrt{x}$
   Domain: $x \geq 0$, because $\sqrt{x}$ requires $x \geq 0$.

2. Subtraction:
   $(f - g)(x) = (x^2 + 4) - \sqrt{x}$
   Domain: $x \geq 0$, same reason as above.

3. Multiplication:
   $(f \cdot g)(x) = (x^2 + 4) \cdot \sqrt{x}$
   Domain: $x \geq 0$, since $\sqrt{x}$ must be defined.

4. Division:
   $\left(\frac{f}{g}\right)(x) = \frac{x^2 + 4}{\sqrt{x}}$
   Domain: $x > 0$, because $\sqrt{x}$ cannot be zero.

You can follow a similar procedure for the other problems by applying these operations and checking for domains.

Would you like me to continue with the remaining problems or any specific one you find difficult?

5 Related Questions

1. What happens to the domain when both functions contain square roots?
2. How do you handle division by a function with multiple roots (like $g(x) = x^2 - 4$)?
3. What’s the domain if $f(x)$ contains logarithms?
4. How does the domain change for a quotient involving polynomial functions?
5. How do you combine functions if both have restricted domains?

Tip: Always consider the individual domain restrictions from both $f(x)$ and $g(x)$ before combining. The final domain is their intersection!