Final answer:
To find a rational function with specified asymptotes, you can use linear or rational functions. For a horizontal asymptote of y = 3/4 and no vertical asymptote, a linear function like f(x) = (3/4)x would work. For vertical asymptotes at x = 0 and x = 5/2, and a horizontal asymptote at y = -3, a rational function like f(x) = -3(x-5/2)/(x(x-5/2)) can be used.2. Vertical Asymptote: x=0 and x= 5/2
Horizontal asymptote: y= -3
Step-by-step explanation:
To find a rational function with the specified characteristics, we need to consider the horizontal and vertical asymptotes. For the first scenario, where the vertical asymptote is none and the horizontal asymptote is y = 3/4, we can use a simple linear function. One example would be f(x) = (3/4)x. This function has no vertical asymptotes and a horizontal asymptote at y = 3/4.
For the second scenario, where the vertical asymptotes are x = 0 and x = 5/2, and the horizontal asymptote is y = -3, we can use a rational function with linear factors in the denominator. One possible function would be f(x) = -3(x-5/2)/(x(x-5/2)). This function has vertical asymptotes at x = 0 and x = 5/2, and a horizontal asymptote at y = -3.