https://file.aiquickdraw.com/mathbot/file/1727276904222aumw4ujs.jpg

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.

Let f(x) = √(5 + x) and g(x) = √(5 - x). Find the domains of f + g, f - g, f * g, and f / g.

Solve the problem of finding the domains of f + g, f - g, f * g, and f / g, where f(x) = √(5 + x) and g(x) = √(5 - x). This problem involves determining the valid inputs for square root functions and operations between them. Suitable for high school students in Grades 10-12.

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