+0

# 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 $f$

0
55
1
+281

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 $f(x).$

michaelcai  Sep 22, 2017
Sort:

#1
+76929
+1

f(x)   =  [ ax + b ] / [ x + c ]

a = -4      and  -3 + c  = 0 →   c = -3

And    if the x intercept   is (1, 0), then      -4(1)  + b  = 0   →  b  = 4

So

f(x) =   [  -4x + 4 ] /  [ x - 3 ]

See the graph, here :  https://www.desmos.com/calculator/u8akarzdzg

f(x) =  [ rx  +  s ]  / [ 2x + t ]

r =  -8           and       2(3) + t  = 0   →  t  = -6

And  if the x intercept   is (1, 0), then  -8(1)+ s = 0   → s  = 8

So

f (x)   = [ -8x + 8 ] / [ 2x - 6]

See the graph here : https://www.desmos.com/calculator/eeeod2ssah

CPhill  Sep 22, 2017

### 34 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details