Domain and range of a function and interval notation

Functions and graphs: Function machines and composition of functions

Domain and range of a function and interval notation

The domain $D_f$ of a function $f$ is the set of all inputs that a function can accept. The set of all outputs of a function $f$ is called the range $B_f$. To specify the domain or range, you use a set notation where you specify via conditions what properties numbers must have. You often end up with intervals.

Examples

If $f(x)=\sqrt{x}$, then $D_f=B_f=\{x\in\mathbb{R}\mid x\ge 0\}=[0,\infty)$

If $g(x)=\dfrac{1}{x^2+1}$, then $D_g=\mathbb{R}$ and $B_g=\{x\in\mathbb{R}\mid 0<x\le 1\}=(0,1]$

If $h(x)=\dfrac{x}{x+1}$, then $D_h=\{x\in\mathbb{R}\mid x\neq -1\}$ and $B_h=\{x\in\mathbb{R}\mid x\neq 1\}$

What is forbidden? Two forbidden operations in function definitions are "dividing by zero" and "taking the square root of a negative number". In addition, the context can impose a limitation on a domain: For example, if you consider temperature in Kelvin as a quantity, then negative temperatures have no physical meaning.

Interval An interval is a contiguous collection of real numbers, a part of a number line. Below are intervals with the usual notation and the corresponding part of the number line.

Interval Notation	Notation with inequality signs	Part of the number line
\[\ivoo{a}{b}\]	\[a<x<b\]
\[\ivco{a}{b}\]	\[a\le x<b\]
\[\ivoc{a}{b}\]	\[a< x\le b\]
\[\ivcc{a}{b}\]	\[a\le x\le b\]
\[\ivoo{a}{\infty}\]	\[x> a\]
\[\ivco{a}{\infty}\]	\[x\ge a\]
\[\ivoo{-\infty}{b}\]	\[x< b\]
\[\ivoc{-\infty}{b}\]	\[x\le b\]
\[\ivoo{-\infty}{\infty}\]	(all real numbers)

In the interval notation you write the boundary values (the smallest and the largest values, the smallest first) of the interval between two brackets. The shape of the brackets determines whether the boundary value still belongs to the interval or not: A square bracket indicates that the boundary value does belong to the interval, a round bracket indicates that the limit value does not belong to the interval. For intervals that have no boundary on one side, use the infinity symbol and a round bracket. $〈$

The interval notation for all real numbers in the last row of the table above is less commonly used than the $\mathbb{R}$ notation.

Combinations of intervals Intervals can also be combined. For example, all numbers with an absolute value greater than $1$ can also be written as $\ivoo{-\infty}{-1}\cup \ivoo{1}{\infty}$.

Other interval notations Instead of brackets, interval notation also uses the symbols $\langle$ and $\rangle$, so that there can be no misunderstanding with coordinates of points in the plane. For example, you would write $a \le x<b$ as $[a,b\rangle$. In this notation, $x>a$ is also notated as $\langle a,\rightarrow\rangle$, so you don't need the symbol for infinity.

Convert the interval notation $(6,\infty)$ to $\{x\in\mathbb{R}\mid\ldots \}$ replacing the dots with inequalities.

$(6,\infty)$ consists of all numbers $x$ for which $x>6$.

So: $(6,\infty)=\{x\in\mathbb{R}\mid x>6\}$.

New example