Math Tutor - Functions - Theory (2024)

Definition.
The identity function, denoted Id, isdefined for real numbers by

Id(x)=x.

The graph:

This is actually a familiar function, one of the simplest linear functions,so why do we single it out? It has a special meaning if we look at functionsfrom the point of view of algebra and treat them as algebraic objects. Recallthe concept ofcomposition of functions. Ifwe look at the properties of composition as an operation on a set of realfunctions, we find that it has some properties that we know frommultiplication, but lacks others.We know that composition is not commutative, in fact compositionsf○gandg○fare equal only in extremely rare cases, one has to do some work to come upwith such an example. Similarly, there is no hope for the distributive law.

What does work? Composition is associative, that's one thing. Anotherinteresting property is that, and here we are getting to it, it has aunity. By a unity we mean a certain element which, when applied (usingthe operation in question) to an arbitrary element, does not change it.With multiplication it is the number 1. Indeed,1⋅x=x⋅1=xfor any real number x.For composition of functions, the role of unity is played exactly by theidentity function Id. Indeed, for any function f and anyx we have

f○Id: x↦f [Id(x)]=f [x],

which means that f○Id=f, and

Id○f: x↦Id[f (x)] = f (x),

which means that Id○f=fas well. By the way, the second equalityabove follows by realizing that the identity function Id is applied toa certain number f (x), and the action is that it leaves thisnumber as it was.

We just showed that f○Id=Id○f=ffor any real function f,so the identity function really serves as a unity for composition.

For the sake of completeness we remark that when we have a unity, we can askfor an inverse, and just like with multiplication of real numbers, we do havea notion of inverse element for functions and the composition. Indeed, theinverse function f₋₁ is an inverse to f in thealgebraic sense, since the definition of the inverse function can berewritten to meanf₋₁○f=f○f₋₁=Id.We know that notevery function has an inverse, but that is nothing new, because also somereal numbers do not have an inverse with respect to multiplication (namely,x=0 is the culprit).

We actually cheated a bit here. The composition with inverse gives theidentity function only if the function in question has the whole real line asboth its domain and its range. On other cases we have to modify theequalities a bit, which brings us to the next topic.

Identity function on a set

Definition.
Let M be any subset of real numbers. We define itsidentity function Id_M by

Id_M(x)=x for x from M.

So we haveD(Id_M)=R(Id_M)=M.

Note that the identity function we introduced above is just a special case ofthis last definition, it is an identity function on the set of real numbers.Conversely, we can view Id_M as the restriction ofId to the set M.Now we can precisely write the definition of the inverse function:f₋₁○f=Id_D(f)andf○f₋₁=Id_R(f)

Definition.
Let M be any subset of real numbers. We define itscharacteristic function by

The strange X is actually the Greek letter "chi". There is also analternative notation, the characteristic function of a set M isdenoted 1_M. It also has another name, some peoplecall it the indicator function of the set M. A third possiblenotation is I_M, but it is perhaps the worst, sincesome people are lazy to write "d" and use justI_M for the identity function above, so it is easy tomix them up. The first notation is the most widespread, but since Greekletters are a real pain on the Web, I will use in Math Tutor the second one,with boldface 1.

Example: In the picture we show the graph of the characteristicfunction of the indicated set M.

Characteristic functions of sets are very useful, often they are used toexpress "restriction" of a function to some set. Why the quotes? Restrictionmeans that we cut off some pieces of the domain. Here we have a situationwhen we keep the original domain, but we are only interested in the values ofthe function on some part of it, so we make the function zero elsewhere.

Precisely, let f be a function defined on some set M and letN be some subset of M. Let g be the function that hasthe same value as f on N but is zero elsewhere on M.

Then g can be written asg=f⋅1_N.

Indeed, if x is from N, then

[f⋅1_N](x) =f (x)⋅1_N(x) =f (x)⋅1 = f (x),

while for x that is from M but not from N we have

[f⋅1_N](x) =f (x)⋅1_N(x) =f (x)⋅0 = 0.

For another way of expressing characteristic function see the next section onthe Heaviside function.

Heaviside function
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