Real functionsΒΆ

Up until now, we have dealt with sequences – functions from the natural numbers to real numbers. Now we turn our attention to functions from real numbers to real numbers, so-called “real functions”.

A real function is a function from a set, \(A\) to the real numbers, \(\mathbb{R}\). Usually \(A\) will be an “interval” – all numbers between two numbers, all numbers less than a number, all numbers greater than a number or all numbers.

We will usually keep \(A\) implicit, and assume it contains anything we are interested in.

If \(f\) and \(g\) are functions, \(f+g\) is the function defined by \((f+g)(x)=f(x)+g(x)\). Similarly, we define \(fg\), \(-f\) and \(f-g\). If \(f\) is a function, everywhere that \(f(x)\ne 0\), we can define \(1/f\).

We note that with these definitions, functions on a given set \(A\) obey the so-called ring axioms.

The simplest example of a function is \(f(x)=a\), the constant function. The next-simplest example of a function is \(f(x)=x\), the identity function.

With these two functions,

Examples