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)=1/(x+4)

Verified step by step guidance

1. Identify the type of function: The given function $f(x)=\frac{1}{x+4}$ is a rational function.
2. Determine the vertical asymptote: Set the denominator equal to zero, $x+4=0$, which gives $x=-4$.
3. Determine the horizontal asymptote: Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$.
4. Identify the domain: The domain is all real numbers except where the denominator is zero, so $x\ne -4$.
5. Match the function with the description: Look for a description that mentions a vertical asymptote at $x=-4$ and a horizontal asymptote at $y=0$.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the quotient of two polynomials. In the case of ƒ(x) = 1/(x+4), the numerator is a constant polynomial (1), and the denominator is a linear polynomial (x + 4). 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 ƒ(x) = 1/(x+4), the vertical asymptote is at x = -4, where the function approaches infinity. Recognizing vertical asymptotes helps in sketching the graph and understanding the function's limits.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For the function ƒ(x) = 1/(x+4), as x becomes very large or very small, the function approaches 0, indicating a horizontal asymptote at y = 0. This concept is crucial for understanding the long-term behavior of rational functions.