Functions - II (Domain and Range , Graphical Approach)

By -
From Previous Post:

 Condition for existence of a Function:

 * Function must be two variable

 * Function must be real valued

 * Function must be single valued

 Single valued means every element belongs to domain has a unique image
or

the lines parallel to y-axis should cut the graph at one point only (vertical line test)

 Let f be a mapping with domain D such that y = f(x) should assume single value for each 'x'.
In the above figure only 1 and 2 are functions

 all the others are not function because for some x ,  y has more than one value

 Domain: The Possible point of x for which we can get finite value of y

 all the previous post of inequality , we have learnt many expression such as root, log , fraction expression, power expression and their respective conditions (condition for which they defined or get real and finite values).

 A Highlight is given below:

 1)If  \frac{{Numerator}}{{Deno\min ator}} ,  denominator \ne 0

 2) if y = \sqrt {f(x)}  , f(x) \ge 0

 3) if  y = {\log _a}x, a \ne 1,a > 0  and x > 0 

 4) if y = f{(x)^{g(x)}} , f(x) \ne 1,f(x) > 0 and g(x) must be defined

 Recall the Steps of solving the inequality (http://iitmathseasy.blogspot.in/2013/03/rulessteps-to-solve-inequalityequality.html)

 In the same way if a function f(x) is defined in some interval or points( by definition domain), and another function g(x) is defined in some other interval and points (by definition domain) .... then sum , subtraction , multiplication and division of the functions are defined on the interval or points which are common to both.

 1) If domain of y = f(x) and y = g(x) are {D_1} and {D_2} respectively then domain of 

 a) y = f(x) \pm g(x) , Domain = {D_1}\bigcap {{D_2}}

b) y = f(x).g(x), Domain = {D_1}\bigcap {{D_2}}

c) y = \frac{{f(x)}}{{g(x)}}, Since here, g(x) \ne 0, therefore Domain = {D_1}\bigcap {{D_2}} - \{ g(x) = 0\} , i.e. we have to eliminate those points from {D_1}\bigcap {{D_2}}  for which g(x) = 0, because denominator \ne 0

 Range: The possible point of y for which we can get finite value of x 

 Range is a bit tricky , way of solution may differ from question to question basis.
Important Note: 

 a) To find out the domain draw lines Perpendicular to x-axis or Parallel to y-axis and find out the interval in which the lines cut or touch the graph (if touch or cut graph anywhere, the point will be in domain).

 (i.e. to find, for which values of 'x' function is defined...
 if for some values of 'x' function is not defined => there is no part of the graph on those points means if you draw a perpendicular line to x-axis through those points => the line will not cut or touch anywhere the graph of function)

 b) To find out the range draw lines Perpendicular to y-axis or Parallel to x-axis and find out the interval( (if touch or cut graph anywhere, the point will be in Range)



Domain and Range are very clear in above figures 1 , 2 and 3.


Comments

  1. erick24 August 2014 at 07:06

    hi, thanks for share this functions, im workking on a cientific calculator for android and this helps a lot !

    ReplyDelete

Post a Comment

Popular posts from this blog

Fractional Part function

By -
Image
10. Fractional Part f(x) = {x}  Every number 'x' can be written as the sum of its integer and fractional parts. x = [x] + {x} Therefore fractional part, {x} = x - [x] For example, 4.7 = 4 + 0.7 or, 0.7 = 4.7 - 4 i.e. fractional part = Number - Integral Part x = 4.7 , [x] = [4.7] = 4 and {x} = {4.7} = 0.7 Consider, {-4.7} {-4.7} = -4.7 - [-4.7] = -4.7 - (-5)  or, {-4.7} = 0.3 if the number is integer, its fractional part is obviously zero. {6} = 6 - [6] = 6 - 6 = 0 Note- fractional part is always positive and it never becomes 1. in other words, fractional part of a number is the difference between the number and its integral part. Remember,  i.e. fractional part is always less than 1 and greater than or equal to zero. When x will be an integer fractional part will become zero i.e. integer has no fractional part. How to draw the graph, note this, , if  because the fractional part is equal to the number if n...
Read more

Properties of Modulus & Argument: Complex Number

By -
Image
Properties of Modulus: Modulus of z is the length of vector representing z form origin to the point z. 1.  If z = x + iy then   2.   &  Since,  ( equality follows when y = 0) and   (equality follows when x = 0) IMP 3.  Since, and,  4.  ,  5.  6.  7.   holds when  i.e. both the vectors are in same direction.  holds when   i.e. both the vectors are in opposite direction In general, 8.  IMP 9.  Explanation: Since,  Therefore,  or,                                                  [Since   (? click here ) ] and  10.     ( Property of Parallelogram ) Properties of Argument: Argument is the angle between the vector representing z from ...
Read more

SIgnum , Modulus and Greatest Integer Functions

By -
Image
7. Signum function , if , if , if Domain = Signum function takes the sign of the x e.g. , Clearly the Signum Function is defined for all Real values i.e. all values of 'x' but it can attain only three values -1,0 and 1. 8. Modulus Function , if , if Domain = R Properties of Modulus: For revision, here is the quick look of the properties of modulus function 1. 2. 3. 4. 9. Greatest integer function f(x) = [x] It is defined as the largest integer less than or equal to 'x' For example, , [5] = 5 , [ -8] = -8 etc. consider 1.3 Step 1 : which integers are less than or equal to 1.3 clearly, .................... -3, -2, -1, 0 and 1 Step 2 : Among the above integers which is greatest clearly it is 1 so, [1.3] = 1 Now, consider -1.3 Step 1: which integers are less than or equal to -1.3 ............ -4, -3 and -2 Step 2 : Among the above integers which is greatest clearly -2 is the greatest so, [-1.3] = -2 c...
Read more