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 $f(x).$

 Aug 23, 2018

"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). "




Horizontal asymptote means set x to infinity so \(f(x)\rightarrow a\)  hence we immediately have \(a=-4\)


Vertical asymptote means set denominator to zero at x = 3, so \(3+c=0 \text{ or } c = -3\)  


x intercept at (1, 0) means \(\frac{a\times1+b}{1+c}=0\)  so  \(\frac{-4+b}{1-3}=0 \text{ or }b = 4\)  


Hence \(f(x)=\frac{-4x+4}{x-3}\text{ or }f(x)=\frac{4(1-x)}{x-3}\) 


Use the same approach for part b.


(Edited to correct silly mistake!  Thanks heureka.)

 Aug 23, 2018
edited by Alan  Aug 23, 2018
edited by Alan  Aug 23, 2018

Why would the horizontal asymptote mean that it it set to infinity and why would it be equal to a and -4? Could you please explain? Sorry, I really don’t understand...,.

Guest Aug 23, 2018


I struggled a little to get my head around it too.   wink



I'm letting y=f(x) some of the time because it is easier for me to work with.


there is a vertical asymptote at x=3 this means that  y tends to =+ or - infinty when x=3

This will happen if the denominator tends to 0 as x tends to 3

hence c=-3


There is a horizontal asymptote at y=-4.

This means that as y tends to  -4,     x  tends to =+ or - infinty 


  \(\displaystyle \lim_{x\rightarrow \infty}\; f(x)=-4\qquad or \qquad \displaystyle \lim_{x\rightarrow -\infty}\; f(x)=-4\\ a=-4\qquad or \qquad a=-4\\ so\;\;a=-4 \)


so we have



\(f(x) = \frac{-4x+b}{x-3}\)


Now sub in  ( 1,0) to get the value of b 

you will get b=4


\(f(x) = \frac{-4x+4}{x-3}\)



If you think about the asymptotes on a graph it might help. It did help me.




And here is the actual graph


Melody  Aug 24, 2018
edited by Melody  Aug 24, 2018

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