Final answer:
To determine the asymptotes of a function, consider the behavior of the function as x approaches positive or negative infinity. Vertical asymptotes occur when the denominator of the function equals zero. Horizontal asymptotes are found by comparing the highest powers of x in the numerator and denominator. Slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator.
Step-by-step explanation:
The given question is asking about the asymptotes of a function. Asymptotes are lines that a graph approaches but never actually touches. The options provided are vertical asymptote, horizontal asymptote, and slant asymptote. To determine the asymptotes of a function, you need to consider the behavior of the function as x approaches positive or negative infinity.
a) Vertical Asymptote: Look for values of x that make the denominator of the function equal to zero. These values will give the vertical asymptotes. For example, if the function is f(x) = (x^2 + 5x)/(x - 3), the vertical asymptote is x = 3 because the denominator x - 3 equals zero at x = 3.
b) Horizontal Asymptote: Look at the highest power of x in the function. If the highest power is the same in the numerator and denominator, divide the coefficients of the highest power terms. The result will give the horizontal asymptote. For example, if the function is f(x) = (3x^2 + 2x)/(2x + 1), the horizontal asymptote is y = (3/2).
c) Slant Asymptote: If the degree of the numerator is exactly one greater than the degree of the denominator, the function may have a slant asymptote. To find the slant asymptote, perform long division or synthetic division. For example, if the function is f(x) = (x^2 - 3x + 5)/(x - 1), the slant asymptote is y = x - 2.
Based on the provided information, you would need to determine which of the asymptotes (if any) apply to the given function.