### 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 \(B_h=\{x\in\mathbb{R}\mid x\neq 1\}\)

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.

\(\ivco{1}{2}\) consists of all numbers \(x\) for which \(1\le x < 2\).

So: \(\ivco{1}{2}=\{x\in\mathbb{R}\mid 1\le x < 2\}\).