To find the domain of the function $f(x) = \frac{7}{\sqrt{x + 2}}$, we need to determine the set of all values of $x$ for which the function is defined.

Step 1: Identify constraints from the square root

The square root function $\sqrt{x + 2}$ is only defined when the expression inside the square root is non-negative. Therefore, we need: $x + 2 \geq 0$ $x \geq -2$ This means $x$ must be greater than or equal to $-2$.

Step 2: Identify constraints from the denominator

The function will be undefined wherever the denominator is zero. For the square root function, $\sqrt{x + 2}$ is zero when: $\sqrt{x + 2} = 0 \implies x + 2 = 0 \implies x = -2$ Since division by zero is undefined, $x = -2$ must be excluded from the domain.

Step 3: Combine the constraints

The function is defined for $x$ in the interval $(-2, \infty)$.

Step 4: Choose the correct answer

The correct answer is: C. $\left( -2, \infty \right)$

This is the domain of the function $f(x)$.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the domain change if the function were $f(x) = \frac{7}{\sqrt{x - 3}}$?
2. What is the range of the function $f(x) = \frac{7}{\sqrt{x + 2}}$?
3. How does the domain of a square root function generally behave?
4. If $f(x) = \frac{7}{x + 2}$, what would the domain be?
5. What is the domain of the function $f(x) = \sqrt{x + 2}$?

Tip: When finding the domain, always check for restrictions caused by square roots and denominators to ensure the function is well-defined.