Questions: Find the domain of the following rational function. R(x)=8(x^2-3x-70)/(9(x^2-100)) The domain is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the domain of the rational function $R(x) = \frac{8(x^2 - 3x - 70)}{9(x^2 - 100)}$, we need to determine the values of $x$ for which the denominator is not zero. The domain will be all real numbers except those that make the denominator zero.

1. Identify the denominator of the function: $9(x^2 - 100)$.
2. Set the denominator equal to zero and solve for $x$: $9(x^2 - 100) = 0$.
3. Solve the equation $x^2 - 100 = 0$ to find the values of $x$ that are not in the domain.
4. Express the domain in interval notation, excluding the values found in step 3.

The given rational function is: $$R(x) = \frac{8(x^2 - 3x - 70)}{9(x^2 - 100)}$$ The denominator of the function is: $$9(x^2 - 100)$$

To find the values of $x$ that are not in the domain, we set the denominator equal to zero: $$9(x^2 - 100) = 0$$

Solving the equation $x^2 - 100 = 0$: $$x^2 - 100 = 0$$ $$x^2 = 100$$ $$x = \pm 10$$ The values $x = -10$ and $x = 10$ make the denominator zero.

The domain of the function is all real numbers except $x = -10$ and $x = 10$. In interval notation, this is: $$(-\infty, -10) \cup (-10, 10) \cup (10, \infty)$$

Final Answer