FUNCTION part 1

Introduction

Definition of a Function

A function from set $A$ to set $B$ is a special relation that maps every member of set $A$ to exactly one member of set $B$.

Characteristics of a function:

1. every member of set $A$ must have a partner,
2. every member of set $A$ has only one partner.

Examples of Functions and Non-Functions

Observe the following figures to distinguish a function from a non-function.

Determining the Number of Possible Functions

Example 1

Determine the number of possible mappings from:

to:

Observe the illustration below.

Since each member of $A$ can be paired with two choices in $B$, the number of possible functions is:

So, the number of possible mappings from $A=\{1,2\}$ to $B=\{a,b\}$ is 4.

Example 2

Determine the number of possible mappings from:

to:

Observe the following figure.

The number of possible functions is:

So, the number of possible mappings from $A=\{1,2,3\}$ to $B=\{a,b\}$ is 8.

Conclusion

If the number of members of set $A$ is $n(A)=a$ and the number of members of set $B$ is $n(B)=b$, then:

1. the number of possible functions from $A$ to $B$ is

2. the number of possible functions from $B$ to $A$ is

Example

The number of possible mappings from:

to:

is …

Known:

• $n(A)=3$
• $n(B)=4$

Asked: the number of possible mappings from $A$ to $B$.

Solution:

Therefore, the number of possible mappings from $A$ to $B$ is 64.

Explanation Video

To better understand this material, watch the following video.

The next topic covers function notation, evaluating functions, determining a function rule from known values, and one-to-one correspondence. That material is presented in Function part 2.