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

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SUMMARY

The range of the function y = 2x/(x - 1) is determined to be (-∞, 2) ∪ (2, ∞) through graphing techniques. The function exhibits a horizontal asymptote at y = 2, indicating that the function approaches this value but never reaches it. The transformation of the function can be expressed as y = 2 + 2/(x - 1), which reveals that the vertical stretch and horizontal translation do not alter the range. The analysis confirms that the graph resembles y = 1/x, modified by a vertical stretch of 2 and a rightward shift of one unit.

PREREQUISITES
  • Understanding of horizontal asymptotes in rational functions
  • Familiarity with function transformations, including vertical stretches and translations
  • Basic graphing skills for rational functions
  • Knowledge of interval notation for expressing ranges
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  • Learn about function transformations in detail
  • Practice graphing rational functions with varying degrees
  • Explore interval notation and its applications in mathematics
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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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