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.