SOLUTION: find the simplest rational function that satisfies the given conditions. vertical asymptotes x=-5, x=4 x intercept (-1,0) horizontal asymptote y=1 please show your work

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Question 1140197: find the simplest rational function that satisfies the given conditions.
vertical asymptotes x=-5, x=4
x intercept (-1,0)
horizontal asymptote y=1
please show your work
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
i chose (x^2 + x - 19) / (x^2 + x - 20).

here's my graph:

x^2 + x - 20 factors out to be (x+5) * (x-4), creating the vertical asymptotes at x = -5 and x = 4.

the horizontal asymptote is as y = 1 because, as positive values of x get larger and larger, and negative values of x get larger and larger, the difference between the numerator and the denominator becomes smaller and smaller, eventually leading to a result that gets closer and closer to 1 but never quite becoming 1 no matter how large the value of x becomes.

the following excel spreadsheet shows the calculations that result in the vertical asymptotes at x = -5 and x = 4, and result in the horizontal asymptotes as plus or minus x gets larger and larger.

Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!

(1) The vertical asymptotes at x=-5 and x=4 mean there must be factors of (x+5) and (x-4) in the denominator.

The x-intercept at (-1,0) means there must be a factor of (x+1) in the numerator.

The simplest rational function that meets those requirements is clearly

Here is a graph:

But that rational function doesn't satisfy the remaining requirement -- that the horizontal asymptote be y=1. The horizontal asymptote of this function is y=0.

For the horizontal asymptote to be y=1, leading term in the numerator has to be equal to the leading term in the denominator. To get that, we need another factor of the form (x-a) in the numerator.

We could choose any additional factor of that form; but the problem asks for the "simplest" rational function that satisfies all the conditions. Since "simplest" is not well defined in this context, we just need to choose some factor of the required form.

One obvious choice would be just "x"; but I chose to use another factor of (x+1), so that there is a double root at x=-1. (And possibly, since the problem showed only one x-intercept, that was the intention....)

So here is a graph of a rational function (and its horizontal asymptote) that meets all the requirements,