Questions: Homework 5.3 - Rational Function Question 2, 5.3.17 Find the domain of the following rational function. H(x)=-8 x^2/(x-7)(x+2) 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 \( H(x) = \frac{-8x^2}{(x-7)(x+2)} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined where the denominator is zero. Therefore, we need to find the values of \( x \) that make \( (x-7)(x+2) = 0 \).

1. Identify the values of \( x \) that make the denominator zero.
2. Exclude these values from the domain.
3. Express the domain in interval notation.

To find the domain of the rational function \( H(x) = \frac{-8x^2}{(x-7)(x+2)} \), we first identify the values of \( x \) that make the denominator zero. The denominator is \((x-7)(x+2)\).

We solve the equation \((x-7)(x+2) = 0\) to find the values of \( x \) that make the denominator zero: \[ (x-7)(x+2) = 0 \] \[ x - 7 = 0 \quad \text{or} \quad x + 2 = 0 \] \[ x = 7 \quad \text{or} \quad x = -2 \]

The function \( H(x) \) is undefined at \( x = 7 \) and \( x = -2 \). Therefore, we exclude these values from the domain.

The domain of \( H(x) \) is all real numbers except \( x = 7 \) and \( x = -2 \). In interval notation, this is expressed as: \[ (-\infty, -2) \cup (-2, 7) \cup (7, \infty) \]

Final Answer