Questions: Find the equation of the Domain in Interval Notation and the equation of the slant asymptote of the rational function. F(x) = (24x^2 - 6x + 3) / (6x - 3)

Solution Steps

To find the domain of the rational function, identify the values of \( x \) that make the denominator zero, as these are the points where the function is undefined. For the slant asymptote, perform polynomial long division on the numerator and the denominator since the degree of the numerator is one more than the degree of the denominator.

To find the domain of the function \( F(x) = \frac{24x^2 - 6x + 3}{6x - 3} \), we need to identify the values of \( x \) that make the denominator zero, as these are the points where the function is undefined. Solving the equation \( 6x - 3 = 0 \) gives:

\[ x = \frac{1}{2} \]

Thus, the domain of the function in interval notation is all real numbers except \( x = \frac{1}{2} \), which can be expressed as:

\[ (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty) \]

Since the degree of the numerator (2) is one more than the degree of the denominator (1), there is a slant asymptote. To find it, we perform polynomial long division of the numerator by the denominator:

\[ \frac{24x^2 - 6x + 3}{6x - 3} = 4x + 1 \]

The quotient \( 4x + 1 \) represents the equation of the slant asymptote.

Final Answer