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 \(f(x) = \frac{ax+b}{x+c}\)
Find an expression for \(f(x)\)
Part (b): Let \(f\) be of the form \(f(x) = \frac{rx+s}{2x+t}\)
Find an expression for \(f(x)\)
+33653 Part (a)
f(x) = (ax + b)/(x + c)
f(infinity) = - 4 so - 4 = a (limit as x goes to infinity of (ax+b)/(x+c) is a) so a = -4
f(3) = infinite so 3 + c = 0 so c = -3
f(1) = 0 so (a + b)/(1 + c) = 0
(-4 + b)/(1 - 3) = 0 so b = 4
hence f(x) = (-4x + 4)/(x - 3) or f(x) = 4(1 - x)/(x - 3)
I'll leave you to do part (b).
