Find an equation of a rational function f that satisfies the

Solution

Given data

vertical asymptote: x = - 2, x = 0
horizontal asymptote: y = 0
x-intercept: 1; f (4) = 1
An equation of a rational function f that satisfies the conditions:

vertical asymptote: x = - 2, x = 0
implies factors of (x+2) and x in the denominator

horizontal asymptote: y = 0
implies the degree of the numerator is less than the degree of the denominator.

x-intercept: 1; f (4) = 1
Implies it passes thru (1,0) and (4,1)

so y = (ax+b)/[x(x+2)}
0 = (a+b)/[1(1+2)}
1 = (4a+b)/[4(4+2)}
simplify
(a + b) = 0
(4a + b) = 24
subtract from each other
a = 8
so the eqn form is
y = (8x-8)/{x(x+2)}
Answer