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For problem 7 in the image you provided, you're asked to find the formulas for \( f/g \) and \( g/f \), as well as to determine the domain of each function.

### Given Functions:
\[ f(x) = \sqrt{1 - x^2} \]
\[ g(x) = \sqrt{1 - x} \]

### 1. Finding \( \frac{f(x)}{g(x)} \):

\[ \frac{f(x)}{g(x)} = \frac{\sqrt{1 - x^2}}{\sqrt{1 - x}} \]

This can be simplified by recognizing that the square roots can be combined:

\[ \frac{f(x)}{g(x)} = \sqrt{\frac{1 - x^2}{1 - x}} \]

### 2. Finding \( \frac{g(x)}{f(x)} \):

\[ \frac{g(x)}{f(x)} = \frac{\sqrt{1 - x}}{\sqrt{1 - x^2}} \]

Similarly, this simplifies to:

\[ \frac{g(x)}{f(x)} = \sqrt{\frac{1 - x}{1 - x^2}} \]

### 3. Determining the Domain:

#### For \( \frac{f(x)}{g(x)} \):
- The expression inside the square roots must be non-negative.
- For \( f(x) = \sqrt{1 - x^2} \), \( 1 - x^2 \geq 0 \), which means \( -1 \leq x \leq 1 \).
- For \( g(x) = \sqrt{1 - x} \), \( 1 - x \geq 0 \), which means \( x \leq 1 \).

Thus, the domain of \( \frac{f(x)}{g(x)} \) is where both conditions are true, i.e., \( -1 \leq x \leq 1 \).

However, there's an additional condition: \( 1 - x \) must not be zero, because that would make the denominator of \( \frac{f(x)}{g(x)} \) zero. Thus, \( x \neq 1 \).

**Domain of \( \frac{f(x)}{g(x)} \):** \( -1 \leq x < 1 \)

#### For \( \frac{g(x)}{f(x)} \):
- Similar reasoning applies as above, with \( x = 1 \) being excluded from the domain.

**Domain of \( \frac{g(x)}{f(x)} \):** \( -1 \leq x < 1 \)

### Summary:
- **Formula for \( \frac{f(x)}{g(x)} \):** \( \sqrt{\frac{1 - x^2}{1 - x}} \)
- **Domain:** \( -1 \leq x < 1 \)
- **Formula for \( \frac{g(x)}{f(x)} \):** \( \sqrt{\frac{1 - x}{1 - x^2}} \)
- **Domain:** \( -1 \leq x < 1 \)

Would you like to go over more details or have any questions? Here are some related questions you might consider:

1. What happens to the function \( \frac{f(x)}{g(x)} \) as \( x \) approaches 1?
2. How would the domain change if the functions involved were slightly altered?
3. Can you express \( \frac{f(x)}{g(x)} \) without a square root, and what would that imply about the domain?
4. What would be the result of \( f(x) + g(x) \) and its domain?
5. How do the graphs of \( f(x) \) and \( g(x) \) behave individually over their domains?

**Tip:** Always check for points where the denominator could be zero when determining the domain of a function involving division.

In Exercises 5-8, find formulas for f/g and g/f. Give the domain of each. (Focus on Exercise 7)

Explore how to find the formulas and domains for the functions f(x) = sqrt(1 - x^2) and g(x) = sqrt(1 - x) in this detailed algebra problem. Suitable for high school students (Grades 9-12).

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