# How to formally write domain of a function?

• Feb 24th 2013, 08:46 PM
jayjay55531
How to formally write domain of a function?
Okay so the domain of f(x) (according to my calculations) is that x<9 and x≠0. How do I write this formally?

"x ∈ (-∞, 0) ∪ (0, 9)"

for interval notation, and

"{x ∈ ℝ | x < 9 | x ≠ 0}"

for set notation?

This is a purely notational question, I'm just wondering what's considered "standard" format/aesthetics-wise in the math world. I don't know if I'm supposed to use the "is an element of" symbol in interval notation and I'm not sure about having two "such that" lines in the set notation?

I know I won't get marked down no matter how I write it on a test, but I'm just curious what's the formalest formal way to do it?
• Feb 24th 2013, 10:04 PM
ibdutt
Re: How to formally write domain of a function?
The first one that is interval notation is generally used in mathematics.
• Feb 24th 2013, 10:32 PM
jayjay55531
Re: How to formally write domain of a function?
Sorry I was unclear the first time, I'm not asking which out of the two is best. What I meant to ask is for each one, is the way I wrote it correct? My doubts are "I don't know if I'm supposed to use the "is an element of" symbol in interval notation and I'm not sure about having two "such that" lines in the set notation?"

Thanks!!
• Feb 25th 2013, 12:39 AM
hollywood
Re: How to formally write domain of a function?
The domain of a function is a set, so you would have just $(-\infty,0)\cup(0,9)$.

The set-builder notation can be a little more flexible - you could write $\{x\in\mathbb{R}|x<9\text{ and }x\ne0\}$ - I think most mathematicians write out "and" or "or" instead of using "&" or "|". I have also seen a colon ":" instead of the vertical bar, so $\{x\in\mathbb{R}:x<9\text{ and }x\ne0\}$. If $x\in\mathbb{R}$ is understood, I have seen $\{x:x<9\text{ and }x\ne0\}$ or even $\{x<9\text{ and }x\ne0\}$, but I'd stay away from those unless you're the professor....

Just about any description is acceptable: $\{x\in\mathbb{R}:\text{x is less than 9 and nonzero}\}$ or $\{x|\text{x is a nonzero real number less than 9}\}$ or even $\{x|x\in(0,9)\text{ or }x\in(-\infty,0)\}$.

- Hollywood
• Feb 25th 2013, 09:30 PM
jayjay55531
Re: How to formally write domain of a function?
Thanks so much! That's exactly what I was looking for.
Btw do you mind clarifying your first sentence? As in, why should you write just (-∞, 0)∪(0, 9) and not x∈(-∞, 0)∪(0, 9)? Thanks again!!!
EDIT: Or I guess what I mean is, how do you define your variable with interval notation? It seems no one ever says "x" anywhere when they say their answer in interval notation.
• Feb 25th 2013, 10:20 PM
fashionsport
Re: How to formally write domain of a function?
Interest rates are almost always given annually regardless of the other time units in the problem. If you want the weekly rate, you have to convert it .

If you show us what you did, we can check your work .
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• Feb 25th 2013, 10:47 PM
hollywood
Re: How to formally write domain of a function?
The first, $(-\infty,0)\cup(0,9)$, defines a set of real numbers, and the second, $x\in(-\infty,0)\cup(0,9)$, is the statement that x is an element of that set.

In the set-builder notation, the x is a variable that is only used within the definition of the set. To state that x is an element of the set, you would have to do something like $x\in\{x\in\mathbb{R}|x<9\text{ and }x\ne0\}$. But this technically uses "x" twice.

- Hollywood