To determine the domain of the function \( f(x) = \frac{2}{x^2 + 12} \), we need to identify all values of \( x \) for which the function is defined. A function with a fraction is undefined when the denominator is zero, as division by zero is undefined.

For this function, the denominator is:

\[
x^2 + 12
\]

1. Set the denominator equal to zero and solve for \( x \):

   \[
   x^2 + 12 = 0
   \]

2. Rearrange to isolate \( x^2 \):

   \[
   x^2 = -12
   \]

3. Since \( x^2 = -12 \) has no real solutions (as no real number squared can equal a negative value), the expression \( x^2 + 12 \neq 0 \) for any real \( x \).

Therefore, there are no restrictions on \( x \) in the real number set, so the domain of \( f(x) = \frac{2}{x^2 + 12} \) is all real numbers:

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

Would you like more details or have any questions about this solution?

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Here are some related questions:
1. What is the range of \( f(x) = \frac{2}{x^2 + 12} \)?
2. How would the domain change if the function were \( \frac{2}{x^2 - 12} \)?
3. How can you determine the domain of functions with square roots in the denominator?
4. What happens to the domain if there are complex solutions to the denominator?
5. Can you find the domain of \( \frac{1}{x^2 + k} \) for any constant \( k \)?

**Tip:** Always check for values that make the denominator zero to find the domain of rational functions.

Explore the steps to determine the domain of the function f(x) = 2/(x^2 + 12). Learn how to find values for which the function is defined, ensuring the denominator is never zero. Ideal for students in Grades 10-12 covering algebra and rational functions.

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