Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=(x²+3x+4)/(x-5)

Verified step by step guidance

1. Identify the given rational function: \(f(x) = \frac{x^2 + 3x + 4}{x - 5}\).
2. Determine the domain by finding values of \(x\) that make the denominator zero. Set the denominator equal to zero: \(x - 5 = 0\).
3. Solve for \(x\) to find the excluded value from the domain: \(x = 5\). So, the domain is all real numbers except \(x = 5\).
4. Analyze the behavior of the function for asymptotes. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is an oblique (slant) asymptote.
5. Find the oblique asymptote by performing polynomial long division of \(x^2 + 3x + 4\) by \(x - 5\) to express \(f(x)\) as a polynomial plus a remainder over \(x - 5\).

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the structure helps analyze its behavior, including domain restrictions and asymptotes.

Domain of a Rational Function

The domain consists of all real numbers except where the denominator equals zero, as division by zero is undefined. For f(x) = (x² + 3x + 4)/(x - 5), x ≠ 5 is excluded from the domain.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes depend on the degrees of numerator and denominator polynomials.