Doug C. answered • 12/09/16

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The x-intercepts lead to the conclusion that the numerator of the rational function must have the factors (x+3) and (x-1).

The fact that there is a vertical asymptote at x=-2 means there must be at least one factor of (x + 2) in the denominator.

In order to have a horizontal asymptote at y = 1 the degree of the numerator must equal the degree of the denominator, so we need to introduce a factor of x into the denominator, but without introducing another vertical asymptote. That leads to the conclusion that another factor of (x + 2) must appear in the denominator.

So far we have f(x) = (x

^{2}+ 2x - 3) / (x^{2}+ 4x + 4)As x approached +/- infinity f(x) approaches 1, so there is the horizontal asymptote. What is f(0)? Looks like -3/4, so all conditions are satisfied. See https://www.desmos.com/calculator/fuurmwqm4g for the graph.