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To find the domain of the function \( f(x) = \frac{x + 13}{x^2 - 100} \), we need to identify the values of \( x \) for which the function is defined. The function is defined when the denominator is not zero.

### Step 1: Set the denominator to zero
The denominator of the function is \( x^2 - 100 \). Set this equal to zero to find the values of \( x \) that cause the function to be undefined:

\[
x^2 - 100 = 0
\]

### Step 2: Solve for \( x \)
Solve the equation:

\[
x^2 = 100
\]

Taking the square root of both sides:

\[
x = \pm 10
\]

### Step 3: Determine the domain
The function is undefined at \( x = 10 \) and \( x = -10 \). Therefore, the domain of the function is all real numbers except these values.

### Domain of \( f(x) \):
\[
\text{Domain: } x \in \mathbb{R}, \; x \neq 10, \; x \neq -10
\]

Or in interval notation:

\[
(-\infty, -10) \cup (-10, 10) \cup (10, \infty)
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the domain of a function with a square root in the denominator?
2. What is the range of the function \( f(x) = \frac{x + 13}{x^2 - 100} \)?
3. How do vertical asymptotes relate to the domain of a rational function?
4. How would you simplify \( \frac{x + 13}{x^2 - 100} \) if possible?
5. What happens to the graph of \( f(x) = \frac{x + 13}{x^2 - 100} \) near \( x = 10 \) and \( x = -10 \)?

**Tip:** Always check where the denominator equals zero when finding the domain of a rational function.

The domain of the function f(x) = (x + 13) / (x^2 - 100)

Learn how to determine the domain of the rational function f(x) = (x + 13) / (x^2 - 100). This problem involves identifying where the denominator equals zero and understanding the domain in terms of real numbers. Suitable for students in Grades 9-11.

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