Find the domain of the function.

​f(x)equals=StartFraction 7 Over StartRoot x plus 2 EndRoot EndFraction7 x+2

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Part 1

Select the correct choice below and fill in the answer box within your choice.

A.

left parenthesis nothing comma nothing right bracketenter your response here,enter your response here 

​(Simplify your​ answers.)

B.

left bracket nothing comma nothing right parenthesisenter your response here,enter your response here 

​(Simplify your​ answers.)

C.

left parenthesis nothing comma nothing right parenthesisenter your response here,enter your response here 

​(Simplify your​ answers.)

D.

left parenthesis negative infinity comma infinity right parenthesis minus StartSet nothing EndSet(−∞,∞)−enter your response here 

​(Simplify your answer. Use a comma to separate answers as​ needed.)

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.

Discover how to find the domain of the function f(x) = 7/sqrt(x + 2) step-by-step. Understand the constraints imposed by square root functions and denominators. Suitable for Grades 11-12.

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