https://file.aiquickdraw.com/mathbot/file/1727066942321fvwyj7f6.jpg

I see the image contains a set of problems, but you've already completed some of them. I'll focus on the unanswered parts:

Let's proceed with solving the problems and finding the domains for the operations between \( f(x) = \sqrt{1 - x} \) and \( g(x) = \sqrt{16 - x^2} \).

### Problem 4: What is the domain of \( f - g \)?
We need to find the domain where both \( f(x) = \sqrt{1 - x} \) and \( g(x) = \sqrt{16 - x^2} \) are defined.

1. For \( f(x) = \sqrt{1 - x} \), the expression inside the square root must be non-negative:
   \[
   1 - x \geq 0 \implies x \leq 1
   \]
   So, \( f(x) \) is defined for \( x \in (-\infty, 1] \).

2. For \( g(x) = \sqrt{16 - x^2} \), the expression inside the square root must be non-negative:
   \[
   16 - x^2 \geq 0 \implies -4 \leq x \leq 4
   \]
   So, \( g(x) \) is defined for \( x \in [-4, 4] \).

To find the domain of \( f - g \), we need the intersection of the domains of \( f(x) \) and \( g(x) \):
\[
x \in (-\infty, 1] \cap [-4, 4] = [-4, 1]
\]
Thus, the domain of \( f - g \) is \( [-4, 1] \).

### Problem 6: What is the domain of \( f \cdot g \)?
This is similar to \( f - g \). The product \( f(x) \cdot g(x) \) is defined where both \( f(x) \) and \( g(x) \) are defined. As calculated before, the intersection of the domains is \( [-4, 1] \).

Thus, the domain of \( f \cdot g \) is \( [-4, 1] \).

### Problem 8: What is the domain of \( \frac{f}{g} \)?
Here, we need to consider two things:
1. Both \( f(x) \) and \( g(x) \) must be defined, which gives the intersection of their domains, \( [-4, 1] \), as previously calculated.
2. Additionally, we need \( g(x) \neq 0 \), since division by zero is undefined.

To find where \( g(x) = 0 \), solve:
\[
\sqrt{16 - x^2} = 0 \implies 16 - x^2 = 0 \implies x = \pm 4
\]
Thus, \( g(x) = 0 \) at \( x = -4 \) and \( x = 4 \).

Therefore, the domain of \( \frac{f}{g} \) is \( [-4, 1] \) excluding \( x = -4 \), which gives:
\[
(-4, 1]
\]

---

If you have any further questions or need additional details, feel free to ask. Here are some related questions to deepen your understanding:

1. How would the domain change if \( f(x) \) were \( \sqrt{x + 1} \)?
2. What is the graphical interpretation of \( f(x) + g(x) \) for these functions?
3. How does the domain of \( f(x) \cdot g(x) \) change if you replace \( g(x) \) with another quadratic function?
4. How would you handle the case if \( g(x) \) had no real roots (i.e., no points where it becomes zero)?
5. What is the effect on the range of \( \frac{f}{g} \) given its domain?

**Tip:** When finding the domain of a function involving square roots or division, always ensure the radicand is non-negative and the denominator is non-zero!

Let f(x) = √(1 - x) and g(x) = √(16 - x²). Find f + g, f - g, f · g, and f/g, and their respective domains.

Learn how to find the domains of function operations including f + g, f - g, f · g, and f / g where f(x) = √(1 - x) and g(x) = √(16 - x²). This problem covers concepts in algebra and function operations, suitable for Grades 10-12.

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