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Part 1
Find the domain of the following rational function.
Upper R left parenthesis x right parenthesis equals StartFraction 5 left parenthesis x squared minus 3 x minus 70 right parenthesis Over 6 left parenthesis x squared minus 100 right parenthesis EndFraction
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Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.
The domain of​ R(x) is StartSet x vertical line x not equals 10 comma x not equals minus 10 EndSet
  
. 
​(Type an inequality in the form xnot equals. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)
B.
The domain of​ R(x) is the set of all real numbers.

We are asked to find the domain of the rational function:

\[
R(x) = \frac{5(x^2 - 3x - 70)}{6(x^2 - 100)}
\]

### Step 1: Identify the restrictions on the domain
The domain of a rational function is all real numbers **except where the denominator is zero**. So, we need to determine where the denominator equals zero and exclude those values from the domain.

The denominator is:

\[
6(x^2 - 100)
\]

To find where the denominator is zero, we set it equal to zero and solve for \(x\):

\[
x^2 - 100 = 0
\]

\[
x^2 = 100
\]

\[
x = \pm 10
\]

Thus, the denominator is zero when \(x = 10\) or \(x = -10\). These values are not allowed in the domain because they would make the function undefined.

### Step 2: Conclusion
The domain of \(R(x)\) consists of all real numbers except \(x = 10\) and \(x = -10\). So, the domain is:

\[
\boxed{A. \ \{ x \mid x \neq 10, x \neq -10 \}}
\]

Would you like any further explanation or details? 

### Related Questions:
1. How do you find the domain of a function with a square root in the denominator?
2. What is the general rule for finding the domain of a rational function?
3. How would the domain change if the function were simplified?
4. Can rational functions have asymptotes related to domain restrictions?
5. What happens to the graph of the function at \(x = 10\) and \(x = -10\)?

### Tip:
Always check both the numerator and denominator for any simplifications after identifying domain restrictions, as cancellations might remove certain discontinuities.

Find the domain of the following rational function: R(x) = (5(x^2 - 3x - 70)) / (6(x^2 - 100)). Select the correct choice: A) x ≠ 10, x ≠ -10, or B) All real numbers.

Learn how to find the domain of a rational function R(x) = (5(x^2 - 3x - 70)) / (6(x^2 - 100)) by identifying where the denominator equals zero. This problem covers factoring quadratic equations and applying domain restrictions in rational functions. Suitable for high school students.

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