U1 Cn2 Functions

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Functions relate dependent and independent variables, where the dependent variable is determined by the independent variable. Functions can be expressed as equations, tables, or graphs. Even functions are symmetric about the y-axis and odd functions are symmetric about the origin. The domain of a function is the set of valid inputs and the range is the set of outputs. A function can be restricted to only certain domain values or composed of multiple functions.

Education Spiritual Technology

Functions and Graphs Unit 1 Class 2 E. Alexander Burt Potomac School

Functions ,[object Object]

The independent variable ,[object Object]

The range of the function is the set of permitted values for the dependent variable.

Set definition of a function ,[object Object]

The same range value can be produced by several domain values. ,[object Object],[object Object]

X = + or – 2 will yield y=4

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(a) Natural Numbers : N = {1,2,3,4,...} (b) Whole Numbers : W = {0,1,2,3,4, } (c) Integer Numbers :  or Z = {...–3,–2,–1, 0,1,2,3, }, Z+ = {1,2,3,....}, Z– = {–1,–2,–3, } Z0 = {± 1, ± 2, ± 3, } (d) Rational Numbers : p Q = { q ; p, q  z, q  0 } (i) R0 : all real numbers except 0 (Zero). (j) Imaginary Numbers : C = {i,, } (k) Prime Numbers : These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers. Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,... (l) Even Numbers : E = {0,2,4,6, } (m) Odd Numbers : O = {1,3,5,7, } Ex. {1, Note : 5 , –10, 105, 3 22 20 7 , 3 , 0 ....} The set of the numbers between any two real numbers is called interval. (a) Close Interval : (i) In rational numbers the digits are repeated after decimal. (ii) 0 (zero) is a rational number. (e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers Ex. { , ,21/3, 51/4,  ,e, } Note: (i) In irrational numbers, digits are not repeated after decimal. (ii)  and e are called special irrational quantities. (iii)  is neither a rational number nor a irrational number. (f) Real Numbers : {x, where x is rational and irrational number} 20 [a,b] = { x, a  x  b } (b) Open Interval: (a, b) or ]a, b[ = { x, a < x < b } (c) Semi open or semi close interval: [a,b[ or [a,b) = {x; a  x < b} ]a,b] or (a,b] = {x ; a < x  b} Let A and B be two given sets and if each element a  A is associated with a unique element b  B under a rule f , then this relation is called function. Here b, is called the image of a and a is called the pre- image of b under f. Note : (i) Every element of A should be associated with Ex. R = { 1,1000, 20/6,  , , –10, – ,.....} 3 B but vice-versa is not essential. (g) Positive Real Numbers: R+ = (0,) (h) Negative Real Numbers : R– = (– ,0) (ii) Every element of A should be associated with a unique (one and only one) element of but any element of B can have two or more relations in A. 3.1 Representation of Function : It can be done by three methods : (a) By Mapping (b) By Algebraic Method (c) In the form of Ordered pairs (A) Mapping : It shows the graphical aspect of the relation of the elements of A with the elements of B . Ex. f1: f2 : f3 : f4 : In the above given mappings rule f1 and f2 shows a function because each element of A is associated with a unique element of B. Whereas f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B. (B) Algebraic Method : It shows the relation between the elem

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U1 Cn2 Functions

1. Functions and Graphs Unit 1 Class 2 E. Alexander Burt Potomac School

2.

3.

4. The range of the function is the set of permitted values for the dependent variable.

5.

6.

7. X = + or – 2 will yield y=4

8.

9. f(x) = p x 2

11.

12. And the vertical (ordinate) axis is the dependent variable (and thus the domain)

13.

14.

15.

16.

17.

18.

20. Evaluate the innermost function first