Functions - 1
Basics and Definitions
A function is a method that takes an input value and gives an output value:
graph LR
x -->|Input| F[Function Machine]
F -->|Output| fx
A function from a set $A$ to a set $B$ is an assignment of exactly one element of $B$ to each element of $A$.
We write $f(a)=b$ if $b$ is the unique element of $B$ assigned by the function $f$ to the element of $a$.
If $f$ is a function from $A$ to $B$ we write $f: A\rightarrow B$.
graph LR
subgraph x
1
2
3
end
subgraph fx
4
5
6
end
1 --> 4
2 --> 5
3 --> 4
6
A function $f:\{1,2,3\} \rightarrow \{4,5,6\}$.
For every value on the left there should be a single value associated to it on the right.
Domain, Co-domain & Range
Suppose $f:A\rightarrow B$
- $A$ is called the domain of $f$.
- $B$ is called the co-domain fo $f$.
- The range $f(A)$ of $f$ is $f(A)=\{f(x)\vert x\in A\}$.
Co-domain v.s. Range
The difference between co-domain and range is that the co-domain is all values in the set $B$ and the range is all the values, $f(x)$, that $A$ maps to via the function $f$.
graph LR
subgraph B
fA
end
A -->|f| fA
The range of $f$.
Example
Give the range of the function:
\[\sin(x):\mathbb{R}\rightarrow\mathbb{R}\]The range of the function would be:
\[\sin(x)=\{x\in\mathbb{R}\vert -1\leq x\leq 1\}\]Composition of Functions
If $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are functions, then their composition $g\circ f$ is a function from $X$ to $Z$ given by:
\[(g\circ f)(x)=g(f(x))\]```mermaid graph LR subgraph X x end subgraph Y subgraph Y’ fx end end subgraph Z gfx end
x –> fx fx –> gfx x –> gfx X –>|f| Y Y –>|g| Z