SOLUTION: 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) = (ax+b)/(x+c). Find an expres

Algebra ->  Finance -> SOLUTION: 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) = (ax+b)/(x+c). Find an expres      Log On


   

Question 1092988: 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) = (ax+b)/(x+c). Find an expression for f(x).
Part (b): Let f be of the form f(x) = (rx+s)/(2x+t). Find an expression for f(x).
Answer by greenestamps(13208) About Me  (Show Source):
You can put this solution on YOUR website!

Part (a): Let f be of the form
f%28x%29+=+%28ax%2Bb%29%2F%28x%2Bc%29

The horizontal asymptote is the ratio of the coefficients of the x term in the numerator and denominator.
a%2F1+=+-4
So a = -4.

The vertical asymptote is where the denominator is 0: at x = 3.
3%2Bc+=+0
So c = -3.

f(1) = 0:
%28-4%2Bb%29%2F%281-3%29=+0
-4%2Bb+=+0

So b = 4.

The completed rational function is
f%28x%29+=+%28-4x%2B4%29%2F%28x-3%29


Part (b): Let f be of the form
f%28x%29+=+%28rx%2Bs%29%2F%282x%2Bt%29

The horizontal asymptote is the ratio of the coefficients of the x term in the numerator and denominator.
r%2F2+=+-4
So r = -8.

The vertical asymptote is where the denominator is 0: at x = 3.
6%2Bt+=+0
So t = -6.

f(1) = 0:
%28-8%2Bb%29%2F%282-6%29=+0
-8%2Bb+=+0

So b = 8.

The completed rational function is
f%28x%29+=+%28-8x%2B8%29%2F%282x-6%29

Note that is in the form required for part (b). However, in simplified form the function is

f%28x%29+=+%28-4x%2B4%29%2F%28x-3%29

which is the same function as in part (a).