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if I have a function $ f(x)=\frac{2x}{x+1}$ I can simplify it to $f(x) =\frac{2}{1}=2$

But this changes the function for example $ f(10)=\frac{20}{11} $ which is not equal to 2. Does this mean if I simplify a function it will become a different function?

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We have $\frac{2x}{x+1} \neq 2$ for all $x \neq -1$. What do you mean by "simplify"? – dtldarek Feb 10 '13 at 15:58
You cannot simplify it in that way. – ciceksiz kakarot Feb 10 '13 at 16:09
Why can't I get rid of the x? – Maik Klein Feb 10 '13 at 16:56

Note that

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

Remember that

$$\frac{a}{b} - c = \frac{a}{b} - \frac{c}{1} = \frac{a - b c}{b} $$

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