What does this, {x|x < 0}, mean?

My question is, "What is the domain and range of this function: y = 1/x ?"

One of the options is Domain: {x|x < 0}, but I don't know what it means to tell if it's right or not.

SurpriseMe  Jul 3, 2017

Domain and range of the function \(y=\frac{1}{x}\)


Now, let's figure out the domain first. The domain of a function just means the possible range of x-values that output a real y-value.


Let's consider that. Is there any number for that would output a nonreal y-value?


Yes, 0. If you plug in 0 for x, you get a nonreal y-value. Will any others? No. I've made a table of the first few domain options.


Possible Domain Corresponding y-value Is it real?
-5 \(\frac{1}{-5}\) Yes
-4 \(\frac{1}{-4}\) Yes
-3 \(\frac{1}{-3}\) Yes
-2 \(\frac{1}{-2}\) Yes
-1 \(\frac{1}{-1}\) Yes
0 \(\frac{1}{0}\) No, a number divided by 0 is undefined.
1 \(\frac{1}{1}\) Yes
2 \(\frac{1}{2}\) Yes
3 \(\frac{1}{3}\) Yes
4 \(\frac{1}{4}\) Yes
5 \(\frac{1}{5}\) Yes


You can see that any numer, except 0, will output a real y-value. Therefore, the domain is any number that isn't 0. 


Domain: \(\{x\in {\rm I\!R}\hspace{1mm}: x\neq0\}\)


In other words, the domain is apart of the real numbers except 0.


Let's think about the range, too. The range is set of all possible values that the function can actually produce. Let's try making a table for that:


Possible Range Corresponding x-value Is it real?
-3 \(-\frac{1}{3}\) Yes
-2 \(-\frac{1}{2}\) Yes
-1 \(-1\) Yes
0 No solution No
1 \(1\) Yes
2 \(\frac{1}{2}\) Yes
3 \(\frac{1}{3}\) Yes


You might be wondering why 0 has "no solution" on it. Well, let's try and solve it:


\(y=\frac{1}{x}\) Substitute in 0 for y.
\(0=\frac{1}{x}\) Multiply my on both sides.
\(0=1\) This is a nonsensical answer. Since \(0\neq1\), there is no solution. Therefore, 0 cannot be a possible range value.


Any other real number works, so our range is:

\(\{f(x)\in {\rm I\!R}\hspace{1mm}:f(x)\neq0\}\)


To recap, the domain and range is as follows


Domain: \(\{x\in {\rm I\!R}\hspace{1mm}: x\neq0\}\)

Range: \(\{f(x)\in {\rm I\!R}\hspace{1mm}:f(x)\neq0\}\)


To answer your second question about what does \(\{x|x\leq0\}\) mean. It means that x is a member of the real numbers such that x is nonnegative (0, 2/9, 1, π, 4.83333..., etc.). However, x has to be nonnegative, so it can't be -3/4, -5, -8.980333333..., etc.

TheXSquaredFactor  Jul 3, 2017

Thank you for your long response! I do actually know how to do the problem, I just didn't understand what that symbol meant in relation to the question. Thanks though.

SurpriseMe  Jul 3, 2017

I think    {x|x < 0}     is read like this:   "x such that x is less than or equal to zero."


You can just read that vertical line like it says, "such that" ...that's how my teacher said it


I finally found a page that agrees with this... here  . This might be a helpful page if you have any more questions about symbols like that.   smiley

hectictar  Jul 3, 2017
edited by hectictar  Jul 3, 2017
edited by hectictar  Jul 3, 2017

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