To find the domain of the function $f(x) = \frac{x^2}{x^2 + 2}$, we need to determine the set of all real values of $x$ for which the function is defined.

Steps to Determine the Domain:

1. Identify the Denominator: The function is defined for all values of $x$ except where the denominator is equal to zero, as division by zero is undefined.

   The denominator of the function is $x^2 + 2$.

2. Set the Denominator Equal to Zero: $x^2 + 2 = 0$ Solve for $x$: $x^2 = -2$ $x = \sqrt{-2}$ Since $\sqrt{-2}$ is not a real number (it's an imaginary number), the equation $x^2 + 2 = 0$ has no real solutions. This means the denominator never equals zero for any real number $x$.

3. Conclusion: Since the denominator $x^2 + 2$ is always positive and never zero for any real number, the function is defined for all real numbers.

Domain in Interval Notation:

The domain of $f(x)$ is all real numbers, which is written in interval notation as: $\text{Domain} = (-\infty, \infty)$

Would you like further details or have any other questions?

Here are five related questions:

1. What is the range of the function $f(x) = \frac{x^2}{x^2 + 2}$?
2. How does the function $f(x) = \frac{x^2}{x^2 + 2}$ behave as $x$ approaches infinity?
3. What are the critical points of the function $f(x) = \frac{x^2}{x^2 + 2}$?
4. How does adding a constant to the denominator affect the domain of a rational function?
5. What is the derivative of $f(x) = \frac{x^2}{x^2 + 2}$?

Tip: Always check if the denominator of a function could potentially equal zero when determining the domain, as this is where a function might be undefined.