6.1 The Basic Definition of a Function

The concept of a function is one that is fundamental to all area of mathematics. We will be using functions thoroughly from now on so it’s a good time to review exactly what a function is and how we can manipulate them.

Definition 6.1.

Let $D$ and $R$ be non-empty sets. Then $f\subseteq D\times R$ is a function if $\forall x\in D,\forall y,z\in R,\ (x,y)\in f$ and $(x,z)\in f\implies y=z.$

In English this says that a function is a binary relation on two sets, and what makes a function different from any old binary relation is the property that every $x\in D$ is paired with a unique $y\in R$ .

Definition 6.2.

Let $f\subseteq D\times R$ be a function. Then we say $f:D\to R$ and denote the unique $y\in R$ associated with $x\in D$ as $f(x)$ .

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$D$ is domain of the function, denoted $\operatorname{dom}(f)$ .
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$R$ is the range¹¹ 1 Some people use the term range in an ambiguous way. A less ambiguous term is codomain, which means the same thing as we have just defined for range. of the function, denoted $\operatorname{range}(f)$ .
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The image of $f$ , $\operatorname{im}(f)$ , is defined as ${\{f(x):x\in D\}}$ , or more explicitly ${\{y\in R:\exists x\in D,f(x)=y\}}$ . The image is a subset of the range, and may or may not be a proper subset.
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For a given $x$ , $f(x)$ may be referred to as the image of $x$ under $f$ , or the value of $f$ at $x$ , or the output of $f$ for the input $x$ .

To express what the image of some function $f$ actually is, we need some kind of formula for the image of an arbitrary point $x\in D$ . This can be given by a straightforward formula such as

f(x)=2x\quad\text{or}\quad x\mapsto x^{2}+1,

(6.1)

or a piecewise definition like

f(x)=\begin{cases}x&x\leq 1\\ x^{2}+1&x>1,\end{cases}

(6.2)

or something even more abstract, such as a power series or recurrence relation (Recall that sequences are defined as functions).

A real function is a function that has real numbers as its inputs and outputs. We can visualise a real functions behaviour by treating the ordered pairs $\big{(}x,f(x)\big{)}\in f$ as Cartesian coordinates in the plane, which is what we all know as a graph.

Example 6.1.

Let $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt{x}$ .
This is not a function since $\sqrt{x}$ is not defined in $\mathbb{R}$ for $x<0$ . Even if we restrict the domain of $f$ to $[0,\infty)$ , it is still technically not a function since there are two possible values of $\sqrt{x}$ for all $x\in[0,\infty)$ . We can fix this by taking the absolute value ( $f:[0,\infty)\to\mathbb{R}$ given by $f(x)=\lvert\sqrt{x}\rvert$ is a function).

Example 6.2.

Let $f:\mathbb{R}\to\mathbb{R}$ be given by

f(x)=\begin{cases}1&\text{if}\ x\in\mathbb{Q}\\ 0&\text{if}\ x\in\mathbb{R}\setminus\mathbb{Q}.\end{cases}

This is again a valid function, and it has numerical values, but drawing a convincing graph of this function is left as a challenge to the reader.