Question

The function is a rational function. Use this function to answer each question. What are the asymptotes of the function? Check all that apply.

Option 1: Horizontal asymptote
Option 2: Vertical asymptote
Option 3: Oblique asymptote
Option 4: No asymptotes

Answer 1

The function y = 1/x has a horizontal asymptote at y = 0 and a vertical asymptote at x = 0.

Step-by-step explanation:

A rational function is a function that can be expressed as the quotient of two polynomial functions. To find the asymptotes of a rational function, we need to analyze the behavior of the function as the input (x) approaches certain values.

1. Horizontal asymptote: If the degrees of the numerator and denominator polynomials are the same, then there is a horizontal asymptote. The horizontal asymptote can be found by comparing the leading coefficients of the numerator and denominator.

2. Vertical asymptote: Vertical asymptotes occur when the denominator becomes zero at certain values of x. To find the x-values where the denominator is zero, set the denominator equal to zero and solve for x.

3. Oblique asymptote: An oblique (slant) asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator. To find the equation of the oblique asymptote, perform long division to divide the numerator by the denominator.

In the case of the function y = 1/x, the degrees of the numerator and denominator are both 1, so there is a horizontal asymptote at y = 0 and a vertical asymptote at x = 0.