Different Types

of Functions

prepared by: Shielamar L. Labiscase

The set of real numbers

SET NOTATION

A set is collection of objects.

The objects in a set are elements or

members of the set.

ROSTER METHOD

of writing a set encloses the elements of the set in braces, {}.

Ex. The set of natural numbers less than 11.

A={1,2,3,4,5,6,7,8,9,10}

A={1,2,3,4,5,6,7,8,9,10}This set has limited number of

elements and is an example of finite set.

To express the fact that 10 is an element of the set, use the symbol , i.e.

10 A.

A second way to denote a set is use to use set-builder notation, where the set is written as

A = { x/x is a natural number less than 11}

Domain of the function

The domain of the function is the set

of consisting of all values of x.

Determine the domain of f(x) =

3/(x+7)

Therefore, the dom(f) is the set of all

real numbers except -7.{x/x=/ -7}

Range of the function

• The range of the functions is the set consisting of all the second components in each element of the set.

–Find the of f(x) = 3/ x+7

• Range (f) = {y/y=/0}

Types of Functions

Constant FunctionIdentity FunctionPolynomial FunctionAbsolute Value

FunctionSquare Root FunctionRational FunctionGreatest Integer

FunctionPiece-wise FunctionUnit Step Function

Constant Functions

The constant function C is function with the range of which is consist of a single number k for all real numbers x in its domain. In symbol C(x) = k.

Graph f(x) = 5 x x -2 -1 0 1 2

y f(x) 5 5 5 5 5

What is the constant function of this graph?

Domain and Range of a Function

• The domain of the constant function is all real numbers

• The range is the constant k. In this function is equal to 5

• The graph is a horizontal line.

Identity Functions

The identity function I is defined by I(x) = x.

The domain is the set of real numbers.

The range of the identity function is also the set of all real numbers.

Graph f(x) = x

X -2 -1 0 1 2y -2 -1 0 1 2

Polynomial Functions

A polynomial in the variable x is a function that can be written in the form,

where an, an-1 , ..., a2, a1, a0 are constants. We call the term containing the highest power of x(i.e. anxn) the leading term, and we call an the leading coefficient. The degree of the polynomial is the power of x in the leading term.

Name of the functionDegree of the

PolynomialName of the

function

0 Constant function

1 Linear function

2 Quadratic function

3 Cubic Function

4 Quartic Function

5 Quintic Function

n (where n > 5) n (where n > 5)

Linear Function

Domain and Range of Linear Function

The domain of a linear function is the set of real numbers {x/x is a real number}.

The range of the linear function is the set of real numbers {y/y is a real number}.

Quadratic function: f(x) = x^2

Domain and Range of Quadratic Function

Domain : {x\x is a real number} Range: If a > 0, {f(x)/f(x) ≥ k}

If a < 0, {f(x)/f(x) ≤ k}

Cubic

Function

Absolute Value Function

It is defined by f(x) = /x/

• The domain of the absolute value function is all real numbers.

• The range is all non negative numbers

Graph of f(x) = /x/Domain: {x/x is a real number}Range: {f(x)/f(x) ≥ 0}

Graph in one Cartesian Coordinate Plane

• y =

• y =

• y = /x/ + 2

• y = /x/ - 2

Graph of y= • To graph y = , simply shift the graph

of y = /x/, A units to the left.• To graph y = , simply shift the graph

of y = /x/, A units to the right.

Graph of y= • To graph y = , simply shift

the graph of y = /x/, B units upward.

• To graph y = , simply shift the graph of y = /x/, B units downward.

Domain: {x/x is a real number}Range: {y/y ≥ 1}

Absolute value function

Domain: {x/x is a real number}Range: {y/y ≤ -2}

Square Root Functions

xxf )(

Domain :Range:Graph:

0xx

0yy

Graph and determine the

domain and range

• h(x) =

• H(x) =

Rational Functions

xxf

1)(

Domain: (x)/x ≠

0}

Range:

{f(x)/f(x) ≠ 0}

Definition of AsymptoteAn asymptote is an imaginary line being approached but never touched or intersected by a graph as it goes through infinity

What is the domain and range of the rational function?

What is the domain and range of the rational

function?

Greatest Integer Function: graph

Domain and range of Greatest Integer

FunctionThe domain of G(x) = [x] is the

set of real numbers.

The range is the set of

integers.

Piece-wise function

• when x is less than 2, it gives x2,• when x is exactly 2 it gives 6• when x is more than 2 and less than or equal

to 6 it gives the line 10-x

It looks like this:

• a solid dot means "including",an open dot means "not including")

Domain and Range of the Piece-wise Function

• The Domain (all the values that can go into the function) is all Real Numbers up to and including 6, which we can write like this

• Dom(f) = (-∞, 6] (using Interval Notation)• Dom(f) = {x | x ≤ 6} (using

Set Builder Notation)

Signum Function

The sign of a real number, also called sgn or signum, is   for a negative number (i.e., one with a minus sign " "), 0 for the number zero, or   for apositive number (i.e., one with a plus sign " ").

In other words, for real  ,

• For real, this can be written

Domain and range of a

Signum function

Domain : {x/x is a set of real

number}

Range : {-1, 0, 1}

Unit Step Function

• It is defined as

• Domain: {x/x is the set of real numbers}

• Range: {0,1}

Thank you for listening.

God bless you all.