The given function is:

\[
f(x) = \sqrt{x^2 + 4} - 8
\]

### Domain:
Since the square root function requires the argument to be non-negative, and \(x^2 + 4 \geq 4\) for all real numbers \(x\), the function is defined for all real numbers. Therefore, the domain of \(f(x)\) is:

\[
\text{Domain of } f(x): (-\infty, \infty)
\]

### Derivative:
Let's find the derivative \(f'(x)\) using the chain rule. 

\[
f(x) = \sqrt{x^2 + 4} - 8 = (x^2 + 4)^{1/2} - 8
\]

Differentiate \(f(x)\) with respect to \(x\):

\[
f'(x) = \frac{1}{2} (x^2 + 4)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + 4}}
\]

So, the derivative is:

\[
f'(x) = \frac{x}{\sqrt{x^2 + 4}}
\]

### Behavior of \(f(x)\) as \(x \to \infty\):
As \(x\) grows large, \(\sqrt{x^2 + 4}\) behaves approximately like \(\sqrt{x^2} = |x|\). So, as \(x \to \infty\):

\[
f(x) \approx |x| - 8
\]

This shows that as \(x\) becomes large, \(f(x)\) increases without bound.

### Range:
Since the square

Explore the domain, derivative, and behavior of the function f(x) = √(x^2 + 4) - 8. Learn how to calculate the derivative using the chain rule and analyze the function's behavior as x approaches infinity. Suitable for advanced high school and college-level calculus students.

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