please help me. Determine, with support, a rational please help me .function of the form f(x) = g(x)/h(x) that has the following features. Justify your thinking for full marks.

Question

 x − intercept: x =2/3

 y − intercept: y = −4

 point discontinuity (a hole) at (4, 4)

 a horizontal asymptote at y = 6

 a vertical asymptote at x = −1

Answer 1

Hi Zhuuin Z.,

Quite a list of requirements, eh?

Consider this problem step-by-step; you could assemble the forms of g(x) and h(x) which will do all these things, finding the simplest forms which will do this. Then reconsider (see below!)

The easiest requirement is the vertical asymptote at x=-1: the h(x) expression could have a (x+1) factor in it. Do you see why that produces an asymptote there?

The x-intercept of 2/3 is the next thing to pull in: you could put an (x-(2/3)) factor in the g(x) expression. Do you see why that forces y=0 (hence, an x-intercept)?

Now things get more challenging. Two requirements (y-intercept -4, and hole at (4,4)) are satisfied by arbitrary manipulation of numerator function degree (i.e. message a polynomial until it yields the desired values) and a specific knockout multiplier ((x^2-4)/(x^2-4)) respectively.

But the horizontal asymptote is tougher: possibly satisfied as an exponential approach, (perhaps).

Now, all this would provide the vertical asymptote, all right, and function on x to either side. But, to re-consider, if you didn't care if the function was defined across the entire domain of x (except for x=4), you could slip by with something like a y=1/(x^2) type function shape, flipped over the x-axis, shifted up and left, and scaled to fit the other specific points, etc. Note that that would have both x and y asymptotes, automatically! Note also that the "x" in that function can be practically anything, as long as the argument is "0" when at the required asymptote position.

Note that this problem also does not require that g(x) and h(x) have polynomial forms, nor that they are continuous! -- they can have arbitrary complexity. So if you wanted to, say, multiply the numerator by (x^2-4)/(x^2-4) to give the knockout at x=4, you could do it. You could even cheat and specify domain special points for the functions (i.e. if x=4, then f(x) = 20, else f(x) = .....). But that might be a bit "over the top"?

So, you could play around with this problem from various design approaches. For example, you could introduce additional asymptotes, if that helps you manipulate your f values.

It might also help to sketch the various requirements, so that you may more easily visualize how to poke the functions around.

I'd be curious to see what you come up with!

-- Cheers, --Mr. d.