Writing out rules?

Question

I had a really small question that got me wondering today:

Say we have some function $f(x)=x^{-1}$ or say $f(x)=\frac{1}{x}$. So obviously the function is not defined for $x=0$, yet I‘ve rarely seen anyone write out for $x\ne0$.

At the same time no one writes out a bunch of different stuff that would make this function undefined. So I probably answered my own question here but just to make sure, we don‘t have to write these things out right?

Answer 1

Depending on the problem, specific conditions may be stated. In general, the function is only undefined when $x=0$. That is probably why $x>0$ isn't stated. There is no need to state this.

When dealing with rational functions we need to know that the denominator cannot equal zero. So in this case it is obvious that $x \ne 0 $.

Answer 2

Realm of definition sometimes is omitted whenever it is implied "wherever it is defined".

Answer 3

We'll often not bother to write restrictions on domain that are otherwise obvious from the formula used for the function.

That doesn't mean these restrictions don't exist though! One common thing we see is that in certain situations, "obvious" domain restrictions become considerably less obvious. Consider $f(x) = \frac{1}{x}$ and $g(x) = x$. then $h = f\cdot g$ has apparent definition $h(x) = 1$, but because of how it was made, $h$ has a domain of $x \ne 0$, since $f$ has the same restriction, and you'll need to say that out loud because it's no longer implied from the formula.

Answer 4

To be careful, when you specify a function you should specify the domain. Often we neglect that and the domain is supposed to be understood from context. In your example I would assume the domain of $f(x)=\frac 1x$ to be $\Bbb R \setminus \{0\}$ unless the problem only involved positive numbers for some reason. $f(x)=\frac 1x$ for $x \gt 2$ is also a perfectly good function. It is a different function because it has a different domain. There are many places in math where details like this are not written down. Sometimes it leads to confusion. I have seen a number of questions on this site where it was not clear whether the variables were real or integers, for example. If it isn't clear you should spell it out.