Functions

Recall that f : D $\subset$ $\mathbb {R}$$\to$T is a function if f (x) is a well defined value in T for each x $\in$ D. We say that D is the domain of the function, T is the target space and f (D) = {f (x) : x $\in$ D} is the range of f.

Note first that the definition says nothing about a formula; just that the result must be properly defined. And the definition can be complicated; for example

f (x) = $$\left\{\vphantom{\begin{array}{ll} 0&\mbox{if $x \le a$\ or $x \ge b$;}\\ 1&\mbox{if $a<x < b$.} \end{array}}\right.$$$$\begin{array}{ll} 0&\mbox{if $x \le a$\ or $x \ge b$;}\\ 1&\mbox{if $a<x < b$.} \end{array}$$

defines a function on the whole of $\mathbb {R}$, which has the value 1 on the open interval (a, b), and is zero elsewhere [and is usually called the characteristic function of the interval (a, b).]

In the simplest examples, like f (x) = x² the domain of f is the whole of $\mathbb {R}$, but even for relatively simple cases, such as f (x) = $\sqrt{x}$, we need to restrict to a smaller domain, in this case the domain D is {x : x$\ge$0}, since we cannot define the square root of a negative number, at least if we want the function to have real - values, so that T $\subset$ $\mathbb {R}$.

Note that the domain is part of the definition of a function, so changing the domain technically gives a different function. This distinction will start to be important in this course. So f₁ : $\mathbb {R}$$\to$$\mathbb {R}$ defined by f₁(x) = x² and f₂ : [- 2, 2]$\to$$\mathbb {R}$ defined by f₂(x) = x² are formally different functions, even though they both are ``x²'' Note also that the range of f₂ is [0, 4]. This illustrate our first use of intervals. Given f : $\mathbb {R}$$\to$$\mathbb {R}$, we can always restrict the domain of f to an interval I to get a new function. Mostly this is trivial, but sometimes it is useful.

Another natural situation in which we need to be careful of the domain of a function occurs when taking quotients, to avoid dividing by zero. Thus the function

f (x) = $${\frac{{1}}{{x-3}}}$$    has domain {x $\in$ $\mathbb {R}$ : x$\ne$3}.    

The point we have excluded, in the above case 3 is sometimes called a singularity of f.

Exercise 1.9   Write down the natural domain of definition of each of the functions:

f (x) = $${\frac{{x-2}}{{x^2 - 5x + 6}}}$$        g(x) = $${\frac{{1}}{{\sin x}}}$$.

Where do these functions have singularities?

It is often of interest to investigate the behaviour of a function near a singularity. For example if

f (x) = $${\frac{{x-a}}{{x^2-a^2}}}$$ = $${\frac{{x-a}}{{(x-a)(x+a)}}}$$     for x$\ne$a.    

then since x$\ne$a we can cancel to get f (x) = (x + a)^-1. This is of course a different representation of the function, and provides an indication as to how f may be extended through the singularity at a -- by giving it the value (2a)^-1.