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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 o...
ROSTER METHOD 
of writing a set encloses the 
elements of the set in braces, 
{}. 
Ex. The set of natural numbers less 
th...
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...
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 num...
Domain of the function 
The domain of the function is the set of 
consisting of all values of x. 
Determine the domain o...
Range of the function 
• The range of the functions is the set 
consisting of all the second 
components in each element o...
Constant Function 
 Identity Function 
Polynomial Function 
Absolute Value 
Function 
Square Root Function 
Rational...
Constant Functions 
The constant function C is 
function with the range of which 
is consist of a single number k for 
all...
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....
Identity Functions 
The identity function I 
is defined by I(x) = x. 
The domain is the set 
of real numbers. 
The range o...
Graph f(x) = x 
X -2 -1 0 1 2 
y -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 , ......
Degree of the 
Polynomial 
Name of the function 
0 Constant function 
1 Linear function 
2 Quadratic function 
3 Cubic Fun...
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...
Quadratic function: f(x) = x^2
Domain and Range of Quadratic 
Function 
 Domain : {xx is a real number} 
 Range: If a > 0, {f(x)/f(x) ≥ k} 
If a < 0, {...
Cubic 
Function
Absolute Value Function 
It is defined by f(x) = /x/ 
• The domain of the 
absolute value 
function is all real 
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= 
• To graph y = , simply shift 
the graph of y = /x/, B units 
upward. 
• To graph y = , simply shift 
the gra...
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 
f (x)  x 
Domain : 
Range: 
Graph: 
  0x x 
  0y y
Graph and 
determine the 
domain and range 
• h(x) = 
• H(x) =
Rational Functions 
x 
f x 
1 
( )  
Domain: (x)/x ≠ 0} 
Range: {f(x)/f(x) ≠ 0}
Definition of Asymptote 
An asymptote is an 
imaginary line being 
approached but never 
touched or intersected by 
a grap...
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 Function 
The domain of G(x) = [x] is the set of real 
numbers. 
The range is the set...
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 ...
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 Numb...
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...
• For real, this can be written
{x/x is a set of real number} 
{-1, 0, 1}
• It is defined as 
{x/x is the set of real 
numbers} 
{0,1}
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
Different types of functions
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Different types of functions

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Different types of functions

  1. 1. Different Types of Functions prepared by: Shielamar L. Labiscase
  2. 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. 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. 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. 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. 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. 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. 8. Constant Function  Identity Function Polynomial Function Absolute Value Function Square Root Function Rational Function Greatest Integer Function  Piece-wise Function
  9. 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. 10. Graph f(x) = 5 x x -2 -1 0 1 2 y f(x) 5 5 5 5 5
  11. 11. What is the constant function of this graph?
  12. 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. 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. 14. Graph f(x) = x X -2 -1 0 1 2 y -2 -1 0 1 2
  15. 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. 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. 17. Linear Function
  18. 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. 19. Quadratic function: f(x) = x^2
  20. 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. 21. Cubic Function
  22. 22. 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
  23. 23. Graph of f(x) = /x/ Domain: {x/x is a real number} Range: {f(x)/f(x) ≥ 0}
  24. 24. Graph in one Cartesian Coordinate Plane • y = • y = • y = /x/ + 2 • y = /x/ - 2
  25. 25. 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.
  26. 26. 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.
  27. 27. Domain: {x/x is a real number} Range: {y/y ≥ 1}
  28. 28. Absolute value function
  29. 29. Domain: {x/x is a real number} Range: {y/y ≤ -2}
  30. 30. Square Root Functions f (x)  x Domain : Range: Graph:   0x x   0y y
  31. 31. Graph and determine the domain and range • h(x) = • H(x) =
  32. 32. Rational Functions x f x 1 ( )  Domain: (x)/x ≠ 0} Range: {f(x)/f(x) ≠ 0}
  33. 33. Definition of Asymptote An asymptote is an imaginary line being approached but never touched or intersected by a graph as it goes through infinity
  34. 34. What is the domain and range of the rational function?
  35. 35. What is the domain and range of the rational function?
  36. 36. Greatest Integer Function: graph
  37. 37. 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.
  38. 38. 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
  39. 39. It looks like this: • a solid dot means "including", an open dot means "not including")
  40. 40. 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)
  41. 41. 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 " ").
  42. 42. • For real, this can be written
  43. 43. {x/x is a set of real number} {-1, 0, 1}
  44. 44. • It is defined as {x/x is the set of real numbers} {0,1}

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