given f(x)=√x and g(x)=√4-x^2 find (f/g)(x) then find the domain of f/g

We are given the functions \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt{4 - x^2} \), and we need to find \( \left(\frac{f}{g}\right)(x) \), as well as the domain of \( \left(\frac{f}{g}\right)(x) \).

### Step 1: Finding \( \left(\frac{f}{g}\right)(x) \)
The quotient of \( f(x) \) and \( g(x) \) is given by:

\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{\sqrt{4 - x^2}}.
\]

### Step 2: Finding the domain of \( \left(\frac{f}{g}\right)(x) \)
The domain of \( \left(\frac{f}{g}\right)(x) \) consists of values of \( x \) for which both \( f(x) \) and \( g(x) \) are defined, and \( g(x) \neq 0 \).

#### Domain of \( f(x) = \sqrt{x} \)
The square root function is defined when the argument is non-negative, meaning:
\[
x \geq 0.
\]
So, the domain of \( f(x) \) is \( [0, \infty) \).

#### Domain of \( g(x) = \sqrt{4 - x^2} \)
Similarly, for \( g(x) \) to be defined, the argument inside the square root must be non-negative:
\[
4 - x^2 \geq 0.
\]
This simplifies to:
\[
x^2 \leq 4,
\]
which gives:
\[
-2 \leq x \leq 2.
\]
Thus, the domain of \( g(x) \) is \( [-2, 2] \).

#### Combining the Domains
To find the domain of \( \left(\frac{f}{g}\right)(x) \), we need the intersection of the domains of \( f(x) \) and \( g(x) \), and also ensure that \( g(x) \neq 0 \).

- The domain of \( f(x) \) is \( [0, \infty) \).
- The domain of \( g(x) \) is \( [-2, 2] \).

The intersection of these two domains is \( [0, 2] \), meaning \( x \in [0, 2] \).

#### Ensuring \( g(x) \neq 0 \)
We also need to exclude any values where \( g(x) = 0 \). The function \( g(x) = \sqrt{4 - x^2} \) equals 0 when:
\[
4 - x^2 = 0 \quad \text{or} \quad x = 2.
\]
Thus, \( g(2) = 0 \), and we must exclude \( x = 2 \) from the domain.

### Final Domain
The domain of \( \left(\frac{f}{g}\right)(x) \) is \( [0, 2) \).

### Conclusion
\[
\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{\sqrt{4 - x^2}}
\]
with domain \( x \in [0, 2) \).

Do you need more details or have any questions?

Here are 5 related questions:
1. What is the domain of \( f(x) = \frac{1}{\sqrt{x}} \)?
2. How would the domain change if \( g(x) = \sqrt{9 - x^2} \) instead?
3. Can you find the range of \( f(x) \) and \( g(x) \)?
4. What happens to the domain if we add a constant inside the square roots?
5. How do you find the domain of a general rational function?

**Tip:** Always check if the denominator of a fraction is zero to avoid undefined values when finding domains.

Given f(x) = √x and g(x) = √(4 - x^2), find (f/g)(x) and then find the domain of f/g.

Learn how to find the quotient of functions f(x) = √x and g(x) = √(4 - x^2) and determine its domain. This problem involves understanding the domain of square root functions and rational functions, with a detailed explanation suitable for Grades 10-12.

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