Given the function \( f(x) = \sqrt{4x} - 2x^2 - 5 \), let's break it down:

1. **Square root term**: \( \sqrt{4x} \) is defined only when \( 4x \geq 0 \), or \( x \geq 0 \), since the square root of a negative number is not real.

2. **Polynomial term**: \( -2x^2 - 5 \) is a quadratic function, which is always well-defined for all real numbers.

### Domain of the function:
For \( f(x) \) to be defined, the square root requires \( 4x \geq 0 \), so \( x \geq 0 \).

Thus, the domain of \( f(x) \) is \( [0, \infty) \).

### Simplified expression for \( f(x) \):

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

Would you like further details on solving or analyzing this function?

### Related questions:
1. How do you find the maximum or minimum of the function \( f(x) = 2\sqrt{x} - 2x^2 - 5 \)?
2. What is the derivative of the function \( f(x) \), and how can it help find critical points?
3. How do you determine the concavity of \( f(x) \) using the second derivative?
4. Can you find the x-intercepts of the function \( f(x) \)?
5. What is the range of the function \( f(x) \)?

**Tip:** Always check the domain restrictions when dealing with square root functions.

Explore how to determine the domain and simplify the function f(x) = √(4x) - 2x² - 5. This detailed breakdown covers square roots, quadratic functions, and domain restrictions, ideal for high school students studying algebra and functions.

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