We are given two functions:

$f(x) = \sqrt{5 + x}$ and $g(x) = \sqrt{5 - x}$.

To solve the domain-related problems in the image, we need to determine the domain for each case.

(a) The domain of $f + g$:

The domain of the sum of two functions is the intersection of the domains of $f(x)$ and $g(x)$.

• The domain of $f(x) = \sqrt{5 + x}$: The expression inside the square root, $5 + x$, must be non-negative: $5 + x \geq 0 \implies x \geq -5.$ Therefore, the domain of $f(x)$ is $[-5, \infty)$.

• The domain of $g(x) = \sqrt{5 - x}$: The expression inside the square root, $5 - x$, must also be non-negative: $5 - x \geq 0 \implies x \leq 5.$ Therefore, the domain of $g(x)$ is $(-\infty, 5]$.

The domain of $f + g$ is the intersection of the domains of $f$ and $g$, which is: $[-5, 5].$

(b) The domain of $f - g$:

Similar to part (a), the domain of $f - g$ is the intersection of the domains of $f$ and $g$. Therefore, it remains: $[-5, 5].$

(c) The domain of $f \cdot g$ (product):

Again, the domain of the product of two functions is the intersection of the domains of $f$ and $g$. Thus, the domain of $f \cdot g$ is: $[-5, 5].$

(d) The domain of $\frac{f}{g}$ (quotient):

The domain of the quotient $\frac{f}{g}$ is the intersection of the domains of $f(x)$ and $g(x)$, excluding any points where $g(x) = 0$, since division by zero is undefined.

• We already know that the domain of both $f(x)$ and $g(x)$ is $[-5, 5]$.
• Set $g(x) = 0$ to find where it is undefined: $\sqrt{5 - x} = 0 \implies 5 - x = 0 \implies x = 5.$ Therefore, $g(x) = 0$ when $x = 5$.

Thus, the domain of $\frac{f}{g}$ is $[-5, 5)$ (excluding $x = 5$).

Final Answers:

• (a) The domain of $f + g$ is $[-5, 5]$.
• (b) The domain of $f - g$ is $[-5, 5]$.
• (c) The domain of $f \cdot g$ is $[-5, 5]$.
• (d) The domain of $\frac{f}{g}$ is $[-5, 5)$.

Would you like any further details or clarifications on these results?

Here are five related questions you might consider:

1. How does the square root function impact the domain of a function?
2. What changes would occur in the domain if $f(x)$ and $g(x)$ involved higher powers?
3. What happens to the domain if $g(x)$ were placed in the numerator instead?
4. How would the domains change if $f(x)$ and $g(x)$ were logarithmic functions instead of square root functions?
5. How can we graphically interpret the domains of these functions?

Tip: Always check for points where the denominator equals zero when working with quotient functions to avoid undefined values.