Find the domain of the function.
f left parenthesis x right parenthesis equals StartFraction x squared Over x squared plus 2 EndFraction
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Part 1
The domain is 
  
enter your response here. ​(Type your answer in interval​ notation.)

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.

Find the domain of the function. f(x) = x^2 / (x^2 + 2)

Discover how to find the domain of the rational function f(x) = x^2 / (x^2 + 2). This step-by-step guide covers essential algebraic concepts, including the domain of functions and rational function definitions. Perfect for high school students in Grades 10-12.

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