A function f has a horizontal asymptote of y=-4 a vertical asymptote of x=3 and an x-intercept at (1,0)
Part (a): Let f be of the form
ax+b/x+c
Find an expression for f(x)
Part (b): Let be of the form
rx+s/2x+t
Find an expression for f(x)
+128473 (a) ax + b
______
x + c
If the vetical asymptote = 3 then we want to solve this : 3 + c = 0 → c = -3
If the horizontal asymptote = - 4 then a/1 = -4 → a =-4
Lastly ......if the x intercept is (1, 0) then we need to solve this :
-4(1) + b = 0 → -4 + b = 0 → b = 4
So f(x) = -4x + 4
_________
x - 3
+128473 b) rx + s
______
2x + t
Using the same info
If the vertical asymptote = 3....we need to solve this for t :
2(3) + t = 0 → 6 + t = 0 → t = -6
If the horizontal asymptote = -4, the we need to solve this for r :
r/2 = -4 multiply both sides by 2
r = -8
And since (1,0) is on the graph, then we can solve this for s :
-8 (1) + s = 0
-8 + s = 0
s = 8
So
f(x) = -8x + 8
_______
2x - 6
