How Do You Determine the Range of the Function y = 2x/(x - 1) Through Graphing?

[mathdad]

Find the range of y = 2x/(x - 1) by graphing?

What are the steps? How is this done?

 

[MarkFL]

I would write:

\(\displaystyle y=\frac{2x}{x-1}=\frac{2x-2+2}{x-1}=\frac{2(x-1)+2}{x-1}=2+\frac{2}{x-1}\)

We see this will have a horizontal asymptote at $y=2$, and so the range must be:

\(\displaystyle (-\infty,2)\,\cup\,(2,\infty)\)

We know this will have a graph that is the same as \(\displaystyle y=\frac{1}{x}\), but vertically stretched by a factor of 2 and translated one unit to the right, neither of which affect the range. It will also be translated 2 units up, which will affect the range by moving the horizontal asymptote up 2 units.