A function f has a horizontal asymptote of y = -4, a vertical asymptote of x = 3, and a x-intercept at (1, 0).
Part A: Let f be of the form f(x) = (ax + b)/(x + c). Find an expression for f(x).
Part B: Let f be of the form f(x) = (rx + 3)/(2x + t). Find an expression for f(x).
+9466 Part A
The degree of the numerator is equal to the degree of the denominator, so the horizontal asymptote is
y = a / 1 which we know is -4
a / 1 = -4
a = -4
There is a vertical asymptote at x = 3
So we know when x = 3 , x + c = 0
3 + c = 0
c = -3
Now we know: f(x) = (-4x + b)/(x - 3)
There is an x-intercept at (1, 0) so we know
f(1) = 0
(-4(1) + b)/(1 - 3) = 0
(-4 + b)/( -2 ) = 0
-4 + b = 0
b = 4
Altogether: f(x) = (-4x + 4)/(x - 3)
