Write the equation of a rational function in the form of 𝑓(𝑥) = 𝑎𝑥+𝑏/𝑐𝑥+𝑑

Question

if the vertical asymptote is x =5, the horizontal asymptote is y=2, the x-intercept is (-1/2 , 0) and the y-intercept is (0,-1/5 ).

Answer 1

I am sure you meant:
f(x) = (ax + b)/(cx + d)

"the horizontal asymptote is y=2"
----> ax/(cx) = 2 , as x ---> infinity
a/c = 2
a = 2c
so we have f(x) = (2cx + b)/(cx + d)

"the vertical asymptote is x =5"
---> (x-5) <---> cx + d
c = 1, d = -5

so we have f(x) = (2x + b)/(x - 5)
but x-intercept is (-1/2 , 0)

0 = (-1 + b)/(-1/2 - 5)
b = 1

f(x) = (2x + 1)/(x-5)

check for y-intercept, let x = 0
f(0) = 1/-5 , which agrees with the given.

(the data that the y-intercept was (0,-1/5) was not needed but was a nice way to check my answer)