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For the equation $$f(x) = \dfrac{(3x + |x|)}x$$ how do you algebraically figure out the domain of the function?

I know it's continuous all the way, but I've tried to split the function into $3x+|x|$ which is continuous in all instances of x and then 1/x which is not continuous at zero.

How come this function's domain is still continuous even though 1/x cannot exist at zero?

I feel like I'm missing something really obvious.

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  • $\begingroup$ Does the function have a limit as $x\to 0$? How do you know the domain is continuous? $\endgroup$ – abiessu Jan 8 '14 at 17:12
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    $\begingroup$ What do you mean by "the domain is continous"? How can a set be continous or not? $\endgroup$ – user127.0.0.1 Jan 8 '14 at 17:14
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    $\begingroup$ When you see an absolute value, split it up: for positive values, and negative ones. It's quite helpful. $\endgroup$ – Gigili Jan 8 '14 at 17:14
  • $\begingroup$ @user127001 The concept isn't meaningless, even though there's no property that's called "continuos" for sets, I think either "open", "connected", or "simply connected" may express what he means $\endgroup$ – GPerez Jan 8 '14 at 17:18
  • $\begingroup$ @Shuri what properties do you know about operations on functions, and what happens to their domain? $\endgroup$ – GPerez Jan 8 '14 at 17:20
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Before answering the question, I'd like to dispel what seems to be a misconception. This question about domain has nothing to do with continuity.

You may be confused because in many practical cases, a function is continuous at all points in its domain. For example, this is the case for rational functions. But there are other cases where this is not true.

A function $f$ is defined at a point $a$ when it makes sense to calculate $f(a)$. This is the same as saying that $a$ is in the domain of $f$.

A function $f$ is continuous at a point $a$ if, not only is $f$ defined at $a$, but $\lim_{x \to a} f(x)$ exists and equals $f(a)$.

To solve your problem, you need to answer the following question: for what values of $x$ does the expression $\frac{3x + |x|}{x}$ make sense?

$3x$ always makes sense, and so does $|x|$. Also, you can always add two numbers, no matter what they are. So $3x + |x|$ always makes sense.

The main point, however, is that you can't always divide a number $A$ by a number $B$. When does a fraction $A/B$ make sense?

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Hint: Simplify the function for positive and negative values of $x$.

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$$x>0\longrightarrow f(x)=\frac{3x+|x|}{x}=\frac{3x+\color{red}{x}}{x}=\frac{4x}{x}=4$$ and $$x<0\longrightarrow f(x)=\frac{3x+|x|}{x}=\frac{3x+(\color{blue}{-x})}{x}=\frac{2x}{x}=2$$ and if $x=0$ then ...

enter image description here

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  • $\begingroup$ @J.W.Perry: Oh yes. I did a bad simplification. Fixed and Thanks. $\endgroup$ – mrs Jan 8 '14 at 17:32
  • $\begingroup$ This was a clear answer as well, thank you! $\endgroup$ – Shuri Jan 8 '14 at 17:34
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    $\begingroup$ No problemo there. Good concise answer. $\endgroup$ – J. W. Perry Jan 8 '14 at 17:36

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