Functions - Discrete Mathematics

 

2.101 Introduction to functions

See references 1, 2

Definition:

A function is a relation between a set of inputs A and a set of outputs B.
Each input maps to exactly one output.
Multiple items in A can map to a single item in B.

Every element x in the domain of a function has one output f(x).

Example of a function:

• each item A has an output in B

Example of a relationship that is not a function:

• 65 has no output
• 62 has 2 outputs

$$ f: A \rightarrow B $$

Read as: function f maps A to B.

$$ x \in A: x \rightarrow f(x) = y \ \ \ (y \in B) $$

Definition of domain:

A is the set of inputs and is called the domain of f.
We write:

$$ D_f = A $$

Definition of co-domain:

B is the set of outputs and is called the co-domain of f.
We write:

$$ coD_f = B $$

y is called the image of x,

whereas x is called the pre-image of y.

We write:

$$ f(x) = y $$

Definition of range:

R is the subset of B and a set of all outputs (images) and is called the range of f.

The range of a function is the set of all images.

The range of a function is a subset of its corresponding co-domain.

We write:

$$ R_f \subseteq coD_f $$

Example:

A = {0, 1, 2, 3, 4, ...}

B = {0, 1, 2, 3, 4, ...}

$ x \rightarrow 2x+1 $

$ D_f = A $

$ coD_f = B $

R = {1, 3, 5, 7, 9, ...}

f(0) = 1

f(1) = 3

Image of x, f(x) = y

Exercise:

$ f: Z \rightarrow Z \ with \  f(x) = |x| $ // integers, absolute value of x

$ D_f = Z $

$ coD_f = Z $

$ R_f = Z \{0\} = \{0, 1, 2, 3, ...\} $

f(-1) = f(1) = 1, hence pre-images of 1 = {-1, 1}

Exercise

$ g: R \rightarrow  R \ with \  g(x) = x^2 + 1$ 

$ D_g = R $ // real numbers

$ coD_g = R $

• any number squared is positive
• +1 makes for bigger than 1

$ R_g = [1, +\infty] $

g(-2) = g(2) = 5 hence, pre-images of 5 = {-2, 2}

Plotting Functions

see reference #3

Linear Functions

$$ f(x) = ax + b $$

• straight line function
• passes through the point (0, b)
• a is the gradient

$$ f: R \rightarrow R \ with f(x) = ax + b $$

• if gradient a > 0 then the function is increasing
• $ x_1 < x_2 $ then $ f(x_1) < f(x_2) $

Quadratic Functions

$$ f(x) = ax^2 + bx = c $$

• where a, b, and c are the numbers and a $ \neq $ 0

Exponential Function

$$ f(x)  = b^x \ where \ b > 0 \ and \ b \neq 1 $$

• variable b is called the base of the function
• Domain $ ]-\infty, \infty[ $
• Range $ ]0, \infty[ $
• Horizontal asymptote y= 0

Exponential decay function 

Laws of Exponents

$ b^x \cdot b^y  = b^{x+y} $

$ \frac{b^x}{b^y} = b^{x-y} $

$  (b^x)^y = b^{xy} $

$  (ab)^x = a^x \cdot b^x $

$ (\frac{a}{b})^x = \frac{a^x}{b^x} $

$ b^{-x} = \frac{1}{b^x} $

$ x^{\frac{m}{n}} = ( \sqrt[n]{x})^m  = \sqrt[n]{x^m} $

Natural Logarithm function

see reference 4

2.106 Injective and surjective functions

see reference 5

Injective (one-to-one) Functions

• one-to-one function
• Let $ f: A \rightarrow B $ be a function
• Definition: f is said to be an injective (one-to-one) function if and only if:
• for all $ a,b \in A $, if $ a \neq b $ then $ f(a) \neq f(b) $

Surjective (onto) functions

Resources

1. https://www.coursera.org/learn/uol-discrete-mathematics/lecture/dEASs/2-101-introduction
2. https://www.coursera.org/learn/uol-discrete-mathematics/lecture/dEASs/2-101-introduction
3. https://www.coursera.org/learn/uol-discrete-mathematics/lecture/LJ9wV/2-104-plotting-functions
4. https://www.youtube.com/watch?v=8qs6QxGCIQU
5. https://www.coursera.org/learn/uol-discrete-mathematics/lecture/XyWA9/2-106-injective-and-surjective-functions
6.