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The function given is:

\[
f(x) = \frac{12x}{x^2 + 16}
\]

To find the domain of \( f(x) \), we need to determine where the function is defined. The main restriction for the domain of rational functions (functions that involve a fraction) comes from the denominator: it must not be zero, because division by zero is undefined.

### Step 1: Identify the denominator
The denominator of \( f(x) \) is \( x^2 + 16 \). 

### Step 2: Check where the denominator equals zero
Set the denominator equal to zero and solve for \( x \):

\[
x^2 + 16 = 0
\]

\[
x^2 = -16
\]

Since no real number squared gives a negative result, there are **no real solutions** to this equation. Therefore, the denominator is never zero for any real number.

### Step 3: Conclusion
Since the denominator is never zero for any real value of \( x \), 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:

\[
(-\infty, \infty)
\]

Would you like further explanation or details?

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Here are five related questions to consider:
1. How do you find the domain of a function with square roots or logarithms?
2. What happens if the denominator of a rational function has real solutions?
3. How would the domain change if the denominator were \( x^2 - 16 \)?
4. Can imaginary numbers affect the domain in real-number functions?
5. How does the presence of absolute value functions impact domain restrictions?

**Tip:** Always check the denominator and square roots for restrictions when finding the domain of a function!

Let f(x) = (12x)/(x^2 + 16). Find the domain of f. Write your answer using interval notation.

Learn how to find the domain of the rational function f(x) = 12x / (x^2 + 16). This problem involves understanding the restrictions imposed by the denominator of the function. Suitable for students in Grades 9-11.

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