rational functions?

Question

I am having trouble writing a function with the given conditions. It says to create a rational function with asymptotes at x=4 and y=2 and a domain of {x| x can't equal 4, x can't equal 9}
I think that means something about (x-4) and 2 is the y-intercept or something, I am confused and just need a little help.

Answer 1

Kari, this is a difficult one!

 

For the horizontal asymptote y = 2, we need the numerator and denominator to have the same degree and to be equal to 4 (4/1, 8/2, etc).  This is where y nears but never crosses, so it is not the y-intercept.

 

For the vertical asymptote x = 4, a root of the denominator has to equal 0, something like x-4 = 0, so x = 4.

 

If x can't equal 4 or 9, then we have the same thing, but with 4 and 9, so (x-4)(x-9).  If either x - 4 or x - 9 = 0, then the denominator equals 0.

 

Now, let's combine them all.  Let's start with 4/1.  The denominator is (x-4)(x-9).  Now, this also gives us a vertical asymptote at x = 9, so let's put x - 9 in the numerator to cancel out.

 

So, we have 4(x-9)/(x-9)(x-4).  If we multiply out, though, the numerator's degree is 1 and the denominator's is 2, so we need another x in the numerator.

 

4x(x-9)/(x-9)(x-4) = 4x² - 36 / x² - 13x + 36.

 

Hope this helps!