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

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To determine the range of the function y = 2x/(x - 1) through graphing, the function can be rewritten to identify its behavior. The transformation shows a horizontal asymptote at y = 2, indicating that the range excludes this value. Consequently, the range is defined as (-∞, 2) ∪ (2, ∞). The graph resembles y = 1/x, vertically stretched by a factor of 2 and shifted right and up, but these transformations do not change the overall range. Understanding these characteristics allows for accurate identification of the function's range.
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Find the range of y = 2x/(x - 1) by graphing?

What are the steps? How is this done?
 
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I would write:

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

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

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

We know this will have a graph that is the same as $$y=\frac{1}{x}$$, 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.
 

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