This document defines and provides examples of different types of functions, including: - Constant functions which always output the same number. - Identity functions where the output is equal to the input. - Polynomial functions defined by expressions involving powers of the variable. - Absolute value functions which output the absolute value of the input. - Square root functions which output the square root of the input. - Rational functions where the output is a rational expression involving the input. It also discusses the domain and range of these different function types.

1. Different Types
of Functions
prepared by: Shielamar L. Labiscase

2. 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.

3. 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}

4. 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.

5. 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}

6. 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}

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}

8. Constant Function
 Identity Function
Polynomial Function
Absolute Value
Function
Square Root Function
Rational Function
Greatest Integer
Function
 Piece-wise Function

9. 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.

10. Graph f(x) = 5
x x -2 -1 0 1 2
y f(x) 5 5 5 5 5

11. What is the constant function of this
graph?

12. 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.

13. 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.

14. Graph f(x) = x
X -2 -1 0 1 2
y -2 -1 0 1 2

15. 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.

16. Degree of the
Polynomial
Name 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)

17. Linear Function

18. 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}.

19. Quadratic function: f(x) = x^2

20. Domain and Range of Quadratic
Function
 Domain : {xx is a real number}
 Range: If a > 0, {f(x)/f(x) ≥ k}
If a < 0, {f(x)/f(x) ≤ k}

21. Cubic
Function

23. 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

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

25. Graph in one Cartesian
Coordinate Plane
• y =
• y =
• y = /x/ + 2
• y = /x/ - 2

26. 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.

27. 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.

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

29. Absolute value function

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

31. Square Root Functions
f (x)  x
Domain :
Range:
Graph:
  0x x
  0y y

32. Graph and
determine the
domain and range
• h(x) =
• H(x) =

33. Rational Functions
x
f x
1
( ) 
Domain: (x)/x ≠ 0}
Range: {f(x)/f(x) ≠ 0}

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

35. What is the domain and range
of the rational function?

36. What is the domain and range of the rational function?

38. Greatest Integer Function: graph

39. Domain and range of Greatest Integer Function
The domain of G(x) = [x] is the set of real
numbers.
The range is the set of integers.

40. 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

41. It looks like this:
• a solid dot means "including",
an open dot means "not
including")

42. 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)

43. 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 " ").

44. • For real, this can be written

45. {x/x is a set of real number}
{-1, 0, 1}

46. • It is defined as
{x/x is the set of real
numbers}
{0,1}