Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once.
$\text{ƒ}\left(x\right)=\frac{(x+7)}{(x+1)}$

Verified step by step guidance

1. Identify the given rational function: \(f(x) = \frac{x+7}{x+1}\).
2. Determine the domain by finding values of \(x\) that make the denominator zero. Set the denominator equal to zero: \(x + 1 = 0\).
3. Solve for \(x\) to find the vertical asymptote: \(x = -1\). This value is excluded from the domain.
4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Both numerator and denominator are degree 1, so the horizontal asymptote is the ratio of leading coefficients: \(y = \frac{1}{1} = 1\).
5. Analyze the behavior of the function near the vertical asymptote and at large values of \(x\) to match the function with the correct description in Column II.

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 form and behavior of rational functions is essential for analyzing their properties such as domain, asymptotes, and intercepts.

Domain of a Rational Function

The domain of a rational function includes all real numbers except where the denominator equals zero. Identifying these values is crucial because they create vertical asymptotes or holes in the graph, affecting the function's behavior.

Asymptotes of Rational Functions

Asymptotes are lines that the graph of a rational function approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior, helping to match the function with its description.