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 $f(x).$

Guest Aug 23, 2018

#1**+3 **

"*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).* "

\(f(x)=\frac{ax+b}{x+c}\)

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.)

Alan
Aug 23, 2018

#2**+1 **

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 donâ€™t understand...,.

Guest Aug 23, 2018

#3**+2 **

Hi,

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

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

So

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

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