In this lesson, you will learn about functions in Discrete math, their characteristics, function representations, function types, and examples to better understand the topic.
A function is a rule that assigns exactly one output to each element of a set. For example, if A and B are two non-empty sets, a function f from A to B is an assignment/relation of one element of set B to each element of set A. the function f is donated by f(a) = b.
Example:
draw image slide 21
Note: Functions are also called mappings or transformations.
For f(a) = b, which represents a function from A to B, we say that A is the domain of f, and B is the codomain.
Example: If f(x) = 2x and f: Z ->Z:
the domain and subdomain of f is Z.
the range of f is the set {0, 2, 4, 6, 8 10….}
image slide 22
You can represent a function in three forms:
A function image is the element of the output for a given input. In contrast, the preimage is the input sets’ elements mapped to the output set element.
Example:
Giving the two sets A and B: image in slide 25
If ‘f1’ and ‘f2’ are functions from ‘A’ to ‘B’, then ‘f1’ + f2 and f1 X f2 are also functions of A to B for all x E A, donated by:
(f1 + f2) (x) = f1(x) + f2(x)
(f1 x f2) (x) = f1(x) x f2(x)
Practice
If f1 and f2 are functions from R to R such that f1(x) = 2x and f2(x) = x-1. What are the functions f1+f2 and f1f2?
If f is a function from A to B and S is a subset of A., the image of S under the function f is the subset of B, which is the images of the elements of s. image of S is denoted by f(s).
Example
Consider the two sets, A and B. The function of A to B is as the following
image from slide 26.
What is f(A) and f(b)?
answer , the f(A) = {y,z}
f(B) = {z}
a one-to-one or injunction function is a function that never assigns the same value to two or more different domain elements. In other words, one to one function cannot have two different input values mapped to the same domain element.
one-to-one graph
practice
Is f(x)=x2 and f:N – >N one-to-one?
A onto function, also known as a surjective function, is the function whose every member of the codomain is the image of an element of the domain.
onto function graph
practice:
Is f(x)=x2, where f:N – > N onto?
Is f(x) = x+1 where f:N – >N onto?
a bijective or one-to-one correspondence function is a one-to-one function and onto simultaneously.
graph for the bijective function