Range of Rational Functions

Question

I'm curious how to find the range of rational functions such as y=(x²+6x-10) / (2x-4). I tried the inverse method but ended up getting -y = -19+4x / (x²-4x-9).

Is my method (inverting the equation) correct, or even possible? If not, how would you algebraically prove ranges like this?

Answer 1

First it is improper top-heavy so you divide the polynomials..

(1/2)x+4

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2x - 4 | x^2 + 6x - 10

x^2 - 2x

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8x - 10

8x - 16

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6

So y = (1/2)x+4 + 6/(2x-4)

= (1/2)x +4 + 3/(x-2)

= (1/x)x+4 + e, where 3/(x-2) is an error term

The graph is a line, plus or minus, give or take,

the error term value 3/(x-2)

So with the exception of the asymptote at x=2, the domain

and the range is all reals