11. What are the horizontal and vertical asymptotes for the rational function y = 4x /(x-1)?:
x = 0, y = 0
x = 1, y = 4
x = 4, y = 1
x = 1, y = 0
To find the horizontal and vertical asymptotes for the rational function y = 4x /(x-1), we can analyze the behavior of the function as x approaches infinity or negative infinity, and as x approaches any vertical asymptotes.
Horizontal Asymptote:
When looking for the horizontal asymptote, we want to analyze the behavior of the function as x approaches either positive or negative infinity. To find the horizontal asymptote, we can compare the degrees of the numerator and denominator.
In this case, the degree of the numerator is 1 (x), and the degree of the denominator is also 1 (x-1). Since the degrees are the same, we can find the horizontal asymptote by comparing the coefficients of the highest degree terms.
The coefficient of the highest degree term in the numerator is 4, and the coefficient of the highest degree term in the denominator is 1. Therefore, the horizontal asymptote is given by the division of the coefficients, which is y = 4/1 or y = 4.
Vertical Asymptote:
To find the vertical asymptote, we need to determine the values of x for which the denominator of the rational function becomes zero (which would result in division by zero).
In this case, the denominator is (x-1). To find the vertical asymptote, we set the denominator equal to zero and solve for x:
x - 1 = 0
x = 1
Therefore, the vertical asymptote is x = 1.
So, the horizontal asymptote for the rational function y = 4x /(x-1) is y = 4, and the vertical asymptote is x = 1.