Successfully reported this slideshow.
Upcoming SlideShare
×

# Different types of functions

17,751 views

Published on

by my teacher Ms. Shelamar

Published in: Education
• Full Name
Comment goes here.

Are you sure you want to Yes No
• thank you for sharing Miss Katrina

Are you sure you want to  Yes  No

### 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}