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I mostly tutor community college students ranging from beginning algebra to calculus level. There are several ways in which I explain the domain:

"The domain of a function is all the values you can plug in for $x$ and not get an error, such as dividing by zero."

If given a graph, I will say

"The domain is all the numbers on the $x$- axis above or below which you have a piece of the graph".

Often times this goes ok, but sometime times the students still don't understand. Is there a better way of explaining it?

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    $\begingroup$ And how do you define functions? $\endgroup$ – dtldarek Sep 9 '16 at 9:20
  • $\begingroup$ @dtldarek The question doesn't usually come up, but if it did I would tell them a function is an equation with x and y (or f(x) and x), that when graphed, it doesn't fail the vertical line test. $\endgroup$ – Ovi Sep 9 '16 at 14:15
  • $\begingroup$ Have them shade in the portions of the $x$-axis which are either directly under or directly over the graph. $\endgroup$ – A.Ellett Sep 9 '16 at 22:21
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Actually your definition is not right. Take the function $f:\mathbb R^{+}\to\mathbb R:x\mapsto x^2$ for which the domain is $\mathbb R^{+}$ but there are more $x$ values for which you can evaluate $x^2$. Your method does also not work for all functions. What about complex valued functions or the function which assigns to each person its name?!

I recommend teaching the function as a triple of domain, codomain and an assignment of arguments to its functions value. Take the following functions

  • $f:\mathbb R\to\mathbb R:x\mapsto x^2$
  • $f:\mathbb R\to\mathbb R^{+}_0:x\mapsto x^2$
  • $f:\mathbb R^{+}_0\to\mathbb R:x\mapsto x^2$
  • $f:\mathbb R^{+}_0\to\mathbb R^{+}_0:x\mapsto x^2$

The assignment rule is always the same but the functions differ in their properties (for example injectivity or surjectivity). This example demonstrates that the domain (and codomain) is needed in the definition of the function. So I would say:

A function is an assignment of arguments from a given domain to values of a given codomain.

For motivating the concept you may ask the students about functions in their daily life (like the function which assigns to each person its name). It's important that the students do not undergeneralize the concept of a function.

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    $\begingroup$ The thing you may be overlooking, and it's likely that the Ovi is dealing with, is the convention that if unspecified, the domain for a function $f(x)$ is the largest set of real numbers that results in real number outputs for $f(x)$. See Ratti/McWaters Precalculus (sec 2.4), Sullivan College Algebra, and many others. Ovi's students are most likely having difficulty answering exercises in that vein, which is the crux of his question as I read it. So: If the domain is not given, then how does one come to find it under this convention? $\endgroup$ – Daniel R. Collins Sep 10 '16 at 16:22
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There are a few ways to define the domain of a function that you and your students should consider.

  1. The best and clearest way to specify the domain of a function is by the set of values over which the function is initially defined. Ideally, when your students are given a function, they should be told the domain. As Stephen Kulla explains in his answer, a number of properties like bijectivity, surjectivity, and symmetry depend on the domain over which the function is being considered, and his method for defining the function is the most complete. Another example of functions which are commonly given along with a specified domain are piece-wise functions.

  2. In common high-school and college algebra and precalculus textbooks, such as Larson's Algebra 2 and Blitzer's College Algebra and Precalculus, the domain, when not specified, is considered to be the natural domain of the function. The natural domain is usually considered to be the maximal set over which the function returns real number values. So, when your students find the domain by excluding the values that make the denominator zero or give even-indexed roots of negative numbers, they are really finding the natural domain in this vein.

It is crucial to distinguish between the domains you wish to discuss for a given function. My recommendation is to always use the words natural domain when discussing the domain as in (2) above, because sometimes the domain under discussion will be different from the natural domain. An important example is the sine and cosine functions. In obtaining the arcsine and arccosine functions, we need to consider restricted domains for the sine and cosine functions on which they are bijective, otherwise their inverses are not well-defined. These restricted domains are different from the natural domains, which are both the set of real numbers.

In addition, if one does not distinguish between the maximal set over which the function returns finite values, and that over which the function returns finite real values, additional confusion may occur. I remember being confused why, as a college algebra student, I was required to find complex zeros of a polynomial function but was made to ignore and even avoid complex values of other functions.

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When I'm teaching students who are still struggling to get the idea of domain, I refer to the function as a machine. You throw x's into the machine and it spits out y's. Your domain is x's that the machine understands and can compute. Often times it may be easier to figure out what isn't in your domain (e.g. all reals, except for a few numbers). I would explain this to students as x's that when you throw them into your machine (function), they break the machine. It doesn't know how to process those numbers. These are often divide by 0 issues or square root of negative issues.

Obviously this isn't perfect and doesn't take into account all mathematical functions or domains, but it helps students who struggle with the concept.

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