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To find the domain of the function \( f(x) = \sqrt{18 - 2x} \), we need to ensure that the expression inside the square root is non-negative. This is because the square root of a negative number is undefined in the real number system.

### Step-by-step process:

1. **Set the radicand (expression inside the square root) \(\geq 0\):**
   \[
   18 - 2x \geq 0
   \]
   
2. **Solve the inequality:**
   \[
   18 \geq 2x
   \]
   Divide both sides by 2:
   \[
   9 \geq x
   \]
   or equivalently,
   \[
   x \leq 9
   \]

### Conclusion:
The domain of \( f(x) = \sqrt{18 - 2x} \) is all \( x \) values such that \( x \leq 9 \). In interval notation, this is:

\[
(-\infty, 9]
\]

Would you like a more detailed explanation or have any questions about the process?

Here are some related questions:
1. How would the domain change if the function had been \( f(x) = \sqrt{18 - 3x} \)?
2. What happens to the domain if a square root function includes both a quadratic and linear term inside?
3. Can square roots be negative in complex numbers? What would the domain look like then?
4. How would the range of the function \( f(x) = \sqrt{18 - 2x} \) be determined?
5. If the square root was instead cubed, how would the domain change?

**Tip:** Always check the domain of functions involving square roots by setting the radicand greater than or equal to zero!

Find the domain of the function f(x) = √(18 - 2x). What is the domain of f?

Learn how to find the domain of the square root function f(x) = √(18 - 2x). This step-by-step guide involves solving inequalities to ensure the expression inside the square root is non-negative. Ideal for high school students.

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