EAMCET/ EAPCET SHORT CUT METHODS DAY -25|FUNCTIONS, DOMAIN, RANGE PROBLEMS||SRINIVASA ACADEMY

the question if f of x is a function such that F ofx + y = f ofx + f of Y and F of 1 is = 7 then Sigma r = 1 to n f of R is equals to here there are few functional equations based on the given conditions they are f of x + y = so F ofx into F of Y F ofx into F of Y such function is exponential function f ofx + y = f ofx + f of Y such function is linear function f ofx yal f ofx + f of Y such function is logarithmic function in that way here given that f of x equal to KX F of 1 is = to 7 this implies K into 1 is = 7 K = 7 therefore f of x = 7x summation r = 1 to n f of R will be F of 1 + f of 2 + F of 3 + so on plus F of n f of 1 is 7 F of 2 is 14 F of 3 21 so on F of n is 7 n 7 taken out common 1 + 2 + 3 + 1 till n this is = to 7 into n into n + 1 by 2 7 n into n + 1 by 2 is answer okay very simple question when you know this functional equation so this is the function satisfying the given condition okay see let's move on to question number two f is mapping from R to R be a function defined by F ofx equal to x² - 4x + 3 when X less than 2 x - 3 when X greater than oral 2 then the number of real numbers X for which f of x is equal to 8 the number of real numbers X for Which F of xal to 8 is okay do observe that X greater than or = 2 f of x = 8 implies x - 3 = 8 x = 11 which is possible value because of x = 11 x greater than or equal 2 condition satisfied X less than 2 f of x = 8 implies x² - 4x + 3 = 8 x² - 4x - 5 = 0 x - 5 into x + 1 = 0 x = 5 x = -1 just x = -1 only will satisfies X less than 2 condition we are also getting x equal to 5 when we are writing X is 5 5 S 25 - 4 into 5 20 25 - 20 is 5 + 3 8 we are getting but we are under X less than 2 case so number of such possible values only two this is one of the value this is another value so number of POS possible values only two okay very important question it is a previous year mset question okay next the set of all real values of X for which f of x = to log of 2^ x - 2 to the base 2 plus < TK 1 - x is also real do OB observe that function is real means function is defined function is defined means set of such values of X is called its domain so here there are two functions one is a logarithmic function other one is a square root function logarithmic function defined only when it is real number only when 2^ x - 2 greater than 0 the simpli 2^ X greater than 2^ 1 this implies X greater than 1 okay next square root 1 - x is real only when 1 - x greater than or equal to 0 this implies X less than or equals to 1 okay so simultaneously can you expect the value which is greater than or equal to 1 and Which is less than one is there any real number which is greater than one and same real number is less than or equals to one is it possible no this is real line this is real line somewhere here one greater than one means these numbers will come less than one means these numbers will come there is no common value for x so even though X is equals to 1 here but X strictly greater than one so real number X which is greater than one as well as less than or equals to 1 is impossible greater than one there are infinitely many real numbers less than or equals to 1 there are infinitely many real numbers but same X cannot be greater than one cannot be less than or equals to 1 there is no such real number that's why answer is empty set there is no such real number okay so that's why answer is empty set problem number four that domain of the real valued function f ofx is equal to square root of 2 x² - 7 x + 5 by 3x 2 - 5 x - 2 do observe very carefully and very easy question also square root function is defined means square root function is is real square root function is real only when rootx is defined for X greater than or equal to Z x greater than or equals to Z that's why f of x is defined for 2x² - 7 x + 5 by 3x² - 5x - 2 greater than or equals to 0 which implies factorization 2x² - 2x - 7 x + 5 by 3x² - 6X + x - 2 factorization 2x taken out common 2x taken out common X - 1 minus this is 5X sorry - 5 taken out common X - 1 that's why 2x - 5 denominator 3x - so x - 2 3x Tak out common x - 2 + 1 Tak out common x -2 this is the factorization from here onwards we have to use W curve method 1 2 5x2 is 2.5 - 1X 3 okay so Plus reion or 0 region we need to calculate it is equals to0 at the time of X is equal 1 at the time of X = 5x2 at 1 and 5x2 we have to use closed interval for all the remaining cases open this is positive region however at minus infinity open always uh at - 1x3 also open Union closed 1 comma 2 open then after Union closed 5x 2 comma Infinity open closed 5 by 2 comma Infinity open first option is right answer so w Cur method is important here okay next problem number five the range of the real valued function f ofx equal to modulus x - 2 + modulus x - 3 okay range of the real valued function it occurs minimum or maximum value at xal 2 or xal 3 solution f of x = to modulus x - A + modulus x - B has minimum value at X = A or xal B so that's why minimum value of f ofx minimum value of f of x is f of 2 F of 2 is modulus 2 - 2 + modulus 2 - 3 that is one minimum value is 1 maximum value maximum value at x equal to Infinity only is infinity why because modulus function always greater than or equals to zero so it has no maximum limit that's why it's range is its range is closed interval 1A Infinity open okay so this is the way next the domain of the real valued function f ofx = to square root of 6 x² + 5x - 6 by < TK 4 - x - < TK x + 4 okay domain of the real valued function we know very well that square root function just it is enough to remember square root function defined for X greater than or equals to Z okay solution numerator numerator is defined for 6 x² + 5x - 6 greater than or equal to 0 36 36 mean 6 x² + 9x - 4x - 6 great than or = to 0 this implies 3x taken out common 2x + 3 - 2 taken out common 2x + 3 greater than or equal 0 this implies 2x + 3 into 3x - 2 great than or equals to Z for this V Cur method - 3x2 somewhere here + 2x3 somewhere here plus minus plus greater than or equals to0 happens when - Infinity comma - 3x2 Closed Union closed 2x3 comma Infinity open this is one of the case okay next case square root of 4 - x defined for 4 - x great than or equal to 0 this implies X less than or equal to 4 next square root of x + 4 implies x + 4 great than oral 0 which is X great than or = -4 see on the same real line on the same real line we can also see we can also see - infinity to+ infinity somewhere here - 3x2 this is + 2x3 and uh this is 4 this is is -4 it is giving this domain from the above case and X less than or equal to 4 means this is the reion X greater than or equal to 4 means this is the region now the common case is the common case is - 4 2 - 3x 2 Union 2x 3 to 4 so this is the common case this is the common region so this is right answer it's domain is closed interval -4a - 3x2 okay Union 2x3 comma 4 2x 3A 4 okay so this is the domain of this function let's move on to next question number seven if step X represents the greatest integer less than or equal to X then the range of the real valued function f ofx is equal to 1 by under root step x s + step x - 2 do observe that so first of all f ofx = 1 by root of Step x s + step x - 2 this can also be converted as perfect square like this step x + 1X 2 s always positive as X values are increasing more and more suppose X is tending to Infinity its value is Infinity wherever X is large function has minimum value so that's why this is infinity - 9x4 also tending to Infinity that's why it's that is the maximum value Infinity is the maximum value of denominator function minimum value is zero okay almost minimum value tending to zero not exactly zero and at the same time its domain is required in order to find range step X + 1 by2 s - 9 by 4 strictly positive strictly positive when step X is equal to 2 after that at many places it is positive so in that way in that way minimum value is two in that way minimum value is two minimum value two means maximum can be maximum can be 1 by under root uh 2 + 2 + 1 by 2 whole s - 9 by 4 2 into 2 4 + 1 5 by 2 s 25 by 4 - 9 by 4 16 by 4 that is 1 by < TK4 1 byun 4 is 1x 2 now the range is therefore range is minimum value open interval Z maximum value closed interval 1X 2 so this is the range of the function in order to find range this step is very very important okay is a positive when step X is equal to 2 next question f of x equal to log of x + otk of x² + 1 is an even function an odd function a periodic function neither even nor odd function do observe that do observe that F ofx equal to f of- x it is an even function it is an even function f of - x = - f of x this is an OD function this is an OD function okay so in that way let us try to simplify f of- x that will be log of - x +un x² + 1 which is log of - x +un of x² + 1 by denominator - x - otk x 2 + 1 rationalizing the numerator x² + 1 in the numerator we get a + b into A- B x² - of x² + 1 by - x - under otk x² + 1 gets cancelled Min -1 by this minus gets cancelled we get 1 by X +un x² + 1 which is log of x +un x² + 1 inverse which is - log of x +un x² + 1 which is same as - f of x f of - x is equals to finally we got - f of x that's why it is an odd function okay next the domain and range of f ofx = to cos x - 3 this is f ofx function or respectively very clear F ofx = cos x - 3 3 is a constant function that's why domain of cos x will will be the domain of f ofx so however domain of cos x is capital r in order to find range we know very well that -1 less than or = X less than oral to 1 and -1 - 3 l than or = to cos x - 3 l than or = to 1 - 3 subtracting 3 - 4 L than or = to cos x x - 3 than or = -2 - 4 L than or = f x than or = -2 therefore range is equals to closed interval -4a -2 do observe that domain of your function is capital r range of your function is closed interval - 4 comma -2 so this is the way to find the range of your function we have to to bring F ofx by using suitable operations okay this is the way to find range of the function next question number 10 the domain of the function f ofx is to 1 by S < TK mod x - x so such function is defined only when only when mod x - x greater than zero square root function that is in the denominator actual square root function is defined for inner quantity greater than or equals to Z but it is in the denominator that's why it should be strictly greater than zero this is mod x greater than x mod x greater than x Possible only when X less than Z suppose X less than 0 x value - 5 modulus - 5 greater than - 5 so that's why X less than 0 x less than 0 means it's domain 0 to left hand side that is open interval minus infinity comma 0 x less than 0 open interval - Infinity comma 0 okay this is answer okay so these are the models from functions dear students those who are watching first time my YouTube Channel please do subscribe so that you will get notifications

Channel: Srinivasa Academy

Views: 1,369

Keywords:

Id: uWRfnOv58tM

Length: 24min 14sec (1454 seconds)

Published: Mon May 06 2024