Find the domain of the function. (Enter your answer using interval notation.)

*g*(*x*) = 

3

*x*

*x*2 + 9

To find the domain of the function \( g(x) = \frac{3x}{x^2 + 9} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined wherever the denominator is equal to zero.

### Step 1: Identify the denominator
The denominator of the function is \( x^2 + 9 \).

### Step 2: Determine where the denominator equals zero
We solve the equation:

\[
x^2 + 9 = 0
\]

Subtract 9 from both sides:

\[
x^2 = -9
\]

### Step 3: Analyze the equation
Notice that \( x^2 = -9 \) has no real solutions because the square of any real number cannot be negative. Therefore, the denominator \( x^2 + 9 \) is never zero for any real number \( x \).

### Step 4: Determine the domain
Since the denominator is never zero, the function \( g(x) = \frac{3x}{x^2 + 9} \) is defined for all real numbers \( x \).

### Final Answer:
The domain of \( g(x) \) is:

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

Would you like more details or have any questions?

### Related Questions:
1. What is the range of the function \( g(x) = \frac{3x}{x^2 + 9} \)?
2. Can you find the horizontal asymptote of this function?
3. How would you determine the x-intercepts of the function?
4. What happens to \( g(x) \) as \( x \) approaches infinity?
5. Can you graph the function to confirm the domain and analyze its behavior?

### Tip:
When analyzing the domain of a rational function, always start by checking where the denominator is zero, as these values are not in the domain.

Learn how to find the domain of the function g(x) = 3x / (x^2 + 9) using interval notation. Understand the concept of rational functions and how to determine where a function is defined. Suitable for high school students studying calculus or advanced algebra.

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