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The function given is:

\[
f(x) = 2x^2 + 5\sqrt{x - 2}
\]

To determine the domain of \( f(x) \), we need to find all the values of \( x \) for which the function is defined.

### Step 1: Analyze the quadratic term

The term \( 2x^2 \) is defined for all real numbers. There are no restrictions on \( x \) here.

### Step 2: Analyze the square root term

The square root function \( \sqrt{x - 2} \) is only defined when the expression inside the square root is non-negative. Therefore, we need:

\[
x - 2 \geq 0
\]

Solving this inequality:

\[
x \geq 2
\]

### Conclusion:

The domain of \( f(x) \) is all real numbers \( x \) such that \( x \geq 2 \).

So, the complete statement would be:

"The domain for \( f(x) \) is all real numbers **greater than or equal to 2**."

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Would you like any further details or clarifications?

Here are 5 related questions you might find helpful:
1. How would the domain change if the square root term was \( \sqrt{3 - x} \)?
2. What is the range of the function \( f(x) = 2x^2 + 5\sqrt{x - 2} \)?
3. How do you find the domain of a function involving both square roots and fractions?
4. Can you graph the function \( f(x) = 2x^2 + 5\sqrt{x - 2} \) and interpret its behavior?
5. How does the coefficient of the square root, 5, affect the shape of the graph?

**Tip:** Always ensure that the expression inside a square root is non-negative when finding the domain of functions involving square roots.

If f(x) = 2x^2 + 5√(x - 2), complete the following statement: The domain for f(x) is all real numbers ____ than or equal to 2.

Understand how to determine the domain of the function f(x) = 2x^2 + 5√(x - 2). Learn about the constraints of square root functions and how to identify the values of x for which the function is defined. Suitable for students in Grades 10-12.

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