The equation of a rational function has the form f(x)=(ax+b)/(cx+d). The function has a vertical asymptote at x=-2, horizontal asymptote at y=1, x intercept of 0, y-intercept of 0, and f(x) is positive when x∈(-∞, -2), and negative when x∈(-2,0). What are the values of a, b, c, and d?
The vertical asymptotes in this case happens at the zeros of the denominator
Hence -d/ c = -2 ⇒ d = 2c , d ≠ 0, c;≠ 0.
The horizontal asymptote being 1 means a/c =1 ⇒ a = c.
The fact that the y intercept is 0 this implies f(0) = b/d =0 , and since d≠0 the b= 0.
Therefore thus far f(x)= (ax)/(ax +2a)
f(x) = x/ (x+2 ) which satisfies the rest of the requirements.
