We are given two functions:

$f(x) = \sqrt{x}$ $g(x) = x - 13$

We are asked to find the following operations and their domains:

1. $(f + g)(x)$

This is the sum of the functions $f(x)$ and $g(x)$:

$(f + g)(x) = f(x) + g(x) = \sqrt{x} + (x - 13)$

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

• The function $f(x) = \sqrt{x}$ is defined for $x \geq 0$, because square roots of negative numbers are not real.
• The function $g(x) = x - 13$ is defined for all real values of $x$, since it is a polynomial.

Thus, the domain of $(f + g)(x)$ is determined by the domain of $f(x)$, so:

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

2. $(f - g)(x)$

This is the difference of the functions $f(x)$ and $g(x)$:

$(f - g)(x) = f(x) - g(x) = \sqrt{x} - (x - 13)$

Domain of $(f - g)(x)$:

• Again, $f(x) = \sqrt{x}$ requires $x \geq 0$.
• $g(x) = x - 13$ is defined for all real $x$.

Thus, the domain of $(f - g)(x)$ is also:

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

3. $(f \cdot g)(x)$

This is the product of the functions $f(x)$ and $g(x)$:

$(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x - 13)$

Domain of $(f \cdot g)(x)$:

• $f(x) = \sqrt{x}$ requires $x \geq 0$.
• $g(x) = x - 13$ is defined for all $x$.

Thus, the domain of $(f \cdot g)(x)$ is also:

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

4. $\frac{f}{g}(x)$

This is the quotient of the functions $f(x)$ and $g(x)$:

$\frac{f}{g}(x) = \frac{\sqrt{x}}{x - 13}$

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

• $f(x) = \sqrt{x}$ requires $x \geq 0$.
• $g(x) = x - 13$ is defined for all $x$, but we cannot divide by zero, so we must exclude $x = 13$.

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

$\text{Domain of } \frac{f}{g}(x) = [0, 13) \cup (13, \infty)$

Summary

• $(f + g)(x) = \sqrt{x} + (x - 13)$ has domain $[0, \infty)$
• $(f - g)(x) = \sqrt{x} - (x - 13)$ has domain $[0, \infty)$
• $(f \cdot g)(x) = \sqrt{x} \cdot (x - 13)$ has domain $[0, \infty)$
• $\frac{f}{g}(x) = \frac{\sqrt{x}}{x - 13}$ has domain $[0, 13) \cup (13, \infty)$

Would you like any further clarification or additional questions on these functions?

Here are some related questions to explore:

1. How would you determine the domain of a composite function, such as $(f \circ g)(x)$?
2. What is the difference in domain restrictions between rational and irrational functions?
3. How do you handle square roots of negative numbers in real-valued functions?
4. Can you find the range of the function $f(x) = \sqrt{x}$?
5. How would you solve a real-world problem where square roots and linear functions are combined?

Tip: When working with square roots, always remember that the domain is restricted to values that make the expression under the square root non-negative!