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The function given is \( f(x) = \sqrt{3 - 2x} \). To find the domain of this function, we need to determine the values of \( x \) for which the expression under the square root is non-negative. This ensures that the function produces real values.

### Step 1: Set the expression under the square root greater than or equal to zero
\[
3 - 2x \geq 0
\]

### Step 2: Solve the inequality
\[
-2x \geq -3
\]
\[
x \leq \frac{3}{2}
\]

### Step 3: Write the domain
Since the function is defined for values of \( x \) less than or equal to \( \frac{3}{2} \), the domain is:
\[
(-\infty, \frac{3}{2}]
\]

Thus, the domain of the function \( f(x) = \sqrt{3 - 2x} \) is \( (-\infty, \frac{3}{2}] \).

Would you like further details or have any other questions? Here are some additional related questions to expand your understanding:

1. How would the domain change if the function was \( f(x) = \sqrt{3 + 2x} \)?
2. What is the range of the function \( f(x) = \sqrt{3 - 2x} \)?
3. How would the domain be affected if the square root was squared?
4. What happens to the domain if we replace \( \sqrt{3 - 2x} \) with \( \frac{1}{\sqrt{3 - 2x}} \)?
5. Can you explain how the domain would change if the expression inside the square root were a quadratic?

**Tip**: Always check for non-negative values inside a square root when determining the domain of a function involving radicals!

Find the domain of the function f(x) = sqrt(3 - 2x). Give exact values written as fractions. State your answer using interval notation.

Learn how to find the domain of the function f(x) = sqrt(3 - 2x) by solving the inequality 3 - 2x >= 0. This problem covers key concepts in algebra and domain restrictions for square root functions. Suitable for students in Grades 9-11.

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