Find the domain of f\left(x\right)=\sqrt{-3x-9}

Enter your answer using interval notation.

To find the domain of the function \( f(x) = \sqrt{-3x - 9} \), we need to ensure that the expression inside the square root is non-negative, since the square root of a negative number is not defined in the set of real numbers.

### Step 1: Set the expression inside the square root greater than or equal to zero.
\[
-3x - 9 \geq 0
\]

### Step 2: Solve for \(x\).
First, add 9 to both sides:
\[
-3x \geq 9
\]

Next, divide both sides by -3. Note that when dividing by a negative number, the inequality sign flips:
\[
x \leq -3
\]

### Step 3: Write the domain in interval notation.
Since \(x\) must be less than or equal to \(-3\), the domain is:
\[
(-\infty, -3]
\]

So, the domain of \( f(x) = \sqrt{-3x - 9} \) is \( (-\infty, -3] \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the domain of \( f(x) = \sqrt{4x + 12} \)?
2. How does the domain change for a function \( f(x) = \sqrt{ax + b} \)?
3. What is the domain of \( f(x) = \frac{1}{\sqrt{-2x - 8}} \)?
4. What happens to the domain when you have a cubic root instead of a square root?
5. How do you solve an inequality like \( -5x + 7 \geq 0 \)?

**Tip:** When working with square roots, always ensure the radicand (the expression inside the square root) is non-negative for real-valued functions.

Find the domain of f(x) = √(-3x - 9). Enter your answer using interval notation.

This guide explains how to find the domain of the function f(x) = √(-3x - 9) by solving the inequality -3x - 9 ≥ 0. Step-by-step solution provided, covering algebraic concepts, inequalities, and properties of square roots. Ideal for students in Grades 9-11.

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