Domain

Functions: Functions

Domain

We have previously seen that a function $f:\blue X \to \orange Y$ assigns an element $y$ in a set $\orange Y$ to each element $x$ in the set $\blue X$ . We will now look at these sets in more detail.

Domain and codomain

Let $f$ be a function of $\blue X$ to $\orange Y$ . We call the set $\blue X$ the $\blue{\textbf{domain}}$ of $f$ and the set $\orange Y$ the $\orange{\textbf{codomain}}$ of $f$ .

A function $f$ therefore assigns an element $y$ from the $\orange{\text{codomain}}$ to each element $x$ from the $\blue{\text{domain}}$ .

example

$f: \{1,2,3\} \to \{1,2,\ldots,9\}$

with function rule: $f(x)=x^2$

$\blue{\text{Domain}}$ : $\{1,2,3\}$

$\orange{\text{Codomain}}$ : $\{1,2,3,4,5,6,7,8,9\}$

Not every element reached

In the case of the $\orange{\text{codomain}}$ , it is not necessary that each element is also reached. The part that is reached, is called the range. We will elaborate on this later.

example

$f: \{1,2,3\} \to \{1,2,\ldots,9\}$

with function rule: $f(x)=x^2$

Range: $\{1,4,9\}$

constant function

The constant function is a special function: it is the function that assigns the exact same element $y$ from $\orange{Y}$ to each element $x$ from $\blue{X}$ . In other words, its output is the same for every input value.

The range of this function is the set $\{y\}$ .

Examples

$f(x)=4$

$f(x)=-\frac{1}{3}$

$f(x)=\pi$

Most functions in this course are real-valued functions.

Real-valued and real functions

Functions of which the $\orange{\text{codomain}}$ is $\mathbb{R}$ are called real-valued functions.

Furthermore, if the $\blue{\text{domain}}$ is a contained in $\mathbb{R}$ , we call the function a real function.

When we denote a real function as a function rule it is not immediately clear what the $\blue{\text{domain}}$ is. We say that the $\blue{\text{domain}}$ is the largest possible area in which the function is defined.

Examples

The $\blue{\text{domain}}$ of:

$f(x)=x^2$ is $\blue{\mathbb{R}}$

$f(x)=\sqrt{x}$ is $\blue{\ivco{0}{\infty}}$

$f(x)=\frac{1}{x}$ is $\blue{\mathbb{R}\setminus\{0\}}$

$f(x) = \sqrt{4-x}$ is $\blue{(-\infty, 4]}$

Function rule

This course will often only give a function on the basis of a function rule. The $\blue{\text{domain}}$ is not explicitly stated. In that case, we will assume the largest possible $\blue{\text{domain}}$ in $\mathbb{R}$ . For some types of functions, such as power functions, we will explicitly mention that we will always choose a limited $\blue{\text{domain}}$ .
It is important to take the $\blue{\text{domain}}$ into account, for example, when drawing graphs, or solving equations.

The $\orange{\text{codomain}}$ is often not explicitly mentioned. Throughout this course, we will mainly working with real or real-valued functions, in which case the $\orange{\text{codomain}}$ is equal to $\mathbb{R}$ .

Restricted domain

A real function $f$ represented by a function rule typically has as its domain the greatest possible subset of $\mathbb{R}$ on which the function rule $f(t)$ is defined. We can always choose to limit the domain of $f$ . For example, when working with a function which describes the trajectory of a soccer ball. The function then describes the height of the ball at a time $t$ . If the ball is kicked at time $t=0$ , it only makes sense to limit the domain to $t \geq 0$ .

Explanation example root

Because the domain is the largest possible set on which the function is defined, the domain of the real function $f(x) = \sqrt{x}$ is equal to the set $\ivco{0}{\infty}$ . This is because the root of a negative number is undefined.

Which of the following sets is the largest possible domain of the function $f(x) = {{1}\over{x+5}}$ ?

The interval $\ivoo{-5}{\infty}$ ; the set of all real numbers greater than $-5$

$\ivoo{-\infty}{-5}$ , the set of all real numbers smaller than $-5$

$\mathbb{R}\setminus\{-5\}$ ; the set of all real numbers except $-5$

$RR$ , the set of real numbers