2. Different ways of describing functions

Different ways of describing functions

• 7. Describing functions by recursion

By associations

Let $A=\{a,b,c\}$ and $B=\{1,2,3\}$. We can describe a function $f:A\to B$ by listing all its associations:

• $f(a)=1$
• $f(b)=1$
• $f(c)=2$

We can describe it by using points and arrows:

We can describe the same $f$ by drawing its graph:

A function is a rule, sometimes we can describe it by a single formula, let $A=B=\mathbb{Z}$, for every $x\in \mathbb{Z}$, let:

$$\boxed{f(x)=6x+28}$$

Hence, for example $f(2)=6\times 2+28=40$.

By case distinction

Sometimes the rule can only be described by case distinction, let $A=B=\mathbb{N}$, for every $n\in \mathbb{N}$, let:

$$g(n)=\{\begin{array}{ll}{2}^n,& \text{if }n\text{ is odd}\\ 3n^2+n+1,& \text{if }n\text{ is even}\end{array}$$

Hence, for example $g(4)=3\times 4^2+4+1=53$ and $g(3)=2^3=8$.

Without any formula

The rule may be described without any mathematical formula.

Let $A=\{E|E\text{ is a healthy African elephant}\}$. Let $B=\{e|e\text{ is an elephant ear}\}$.

For every $E\in A$, let $l(E)=E\text{’s left ear}$.

Tuples

From topic 3. Some useful functions.

A tuple can be thought of as a function. For example, the 5-tuple $(22,14,55,1,700)$ can be thought of as a listing of the values of the function $f:\{0,1,2,3,4\}\to \mathbb{N}$ defined by:

$$f(0)=22,f(1)=14,f(2)=5,f(3)=1,f(4)=700$$

Infinite Sequence

An infinite sequence of objects can be thought of as a function.

For example, take $(b_0,b_1,...,b_n)$ is an infinite sequnce of objects from a set $S$. This sequence can be thought of as a listing of the values of the function $f:\mathbb{N}\to S$ defined by:

$$f(n)=b_n$$