Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=(x+7)/(x+1)

Verified step by step guidance

1. Identify the rational function given: \( f(x) = \frac{x+7}{x+1} \).
2. Determine the vertical asymptote by setting the denominator equal to zero: \( x+1=0 \). Solve for \( x \).
3. Find the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both are degree 1, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).
4. Check for any holes in the graph by seeing if there are common factors in the numerator and denominator. In this case, there are none.
5. Analyze the end behavior of the function by considering the horizontal asymptote and the behavior as \( x \to \infty \) and \( x \to -\infty \).

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function ƒ(x)=(x+7)/(x+1), the numerator is a polynomial of degree one, and the denominator is also a polynomial of degree one. Understanding the structure of rational functions is essential for analyzing their behavior, including asymptotes and intercepts.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, leading to undefined values. For the function ƒ(x)=(x+7)/(x+1), setting the denominator x+1 to zero reveals a vertical asymptote at x=-1. This concept is crucial for understanding the limits and behavior of the function near these points.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as x approaches infinity or negative infinity. For the function ƒ(x)=(x+7)/(x+1), the degrees of the numerator and denominator are the same, indicating a horizontal asymptote at y=1, which is found by taking the ratio of the leading coefficients. This concept helps in predicting the long-term behavior of the function.