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