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Functions
:
Let A
B be two non empty sets & F is a relation which associates each elemenet of set A with unique element of set B, then F is c/d a function from A to B.
Set A is called domain of F & B be the co domain of F.
Set of elements of B, which are images of elements of set A is c/d range of F.
F : A
B ("F is function of A into B")
If a
A then element in B which is assigned to 'a' is called image of 'a' & denoted by F(a).
let A = {a, b, c, d}, B = {1, 2, 3, 4, 5}
So, F(a) = 2, F(b) = 3, F(c) = 5, F(d) = 1
Dumb Question
: What is non empty set ?
Ans: A set which contain at least one element
eg. A = {1, 2} is non empty set but B = { } is empty set.
No. of Function (or Mapping) From A to B
:
Let A = {x1, x2, x3, .......... , xm} F : AB& B ={y1, y2, y3, .......... , yn}
Then if each elemenet in set A has n images in set B.
Thus, total no. of functions from A to B = nm
Dumb Question
: How total no. of function from A to B = nm?
Ans: x1
, element in set A take n images x2element in set A take n images ............................................. .............................................
xmcan take n images.
Total no. of function from A to B
 n x n x ............... m times = nm
Domain
: Domain of y = f(x) is set all real x for which f(x) is defined (real).
How to find Domain
:
(i) Expression under even root (i.e. square root, 4throot) 0
(ii) Denominator 0
(iii) If domain of y = (x) & y = y(x) are D1& D2
respectivly then domain of f(x) ± y(x) of f(x).g(x) is D1  D2.
(iv) Domain of is D1 D2- {g(x) = 0}.
Illustration
: Find domain of single valued function y = f(x) given by eq. 10x+ 10y= 10.
Ans: 10
x+ 10y= 10 10y= 10 - 10x y = log10(10 - 10x) {am= b m = logab}
Now, 10 - 10n> 0101> 10x 1 > xdomain x(- , 1)
Dumb Question
: What is single value Function ?
Ans. For every point of domains, these is unique image only.
Range
: Range of y = f(x) is collection of all output & {F(x)} corresponding to each real no. is domain.
How to find range
:
First of all final domain of y = f(x)
(i) If domain
Finite no. of points
range
set of corresponding f(x)values.
(ii) If domainR or R - {some finite points}
Then express x in terms of y. From this find y for x to be defined. (i.e. find values of y for which x exists)
(iii) If domain
a finite interval, find least and greatest value for range using mono tonicity.
Illustration
: Find range of function y = loge(3x2- 4x + 5).
Ans: y is difind if 3x2- 4x + 5 > 0[
Dumb Question
: Why 3x2- 4x + 5 > 0 ?
Ans: ln x is defind only if x > 0]
if x is 0 ln x = -
& for - ve xln is not defind, D = 16 - 4 x 3 x 5 = - 44 < 0& coeff. of x2= 3 > 03x2- 4x + 5 > 0vxR
domainR y = loge(3x2- 4x + 5)3x2- 4x + 5 = ey
3x2- 4x + (5 - ey) = 0Since x is real, So, D0(- 4)2- 4(3)(5 - ey) > 0 ey>
ylog
range[log,)
CLASSIFICATION OF FUNCTIONS
1.
Constant Function
: If range of function f consists of only one no. then f is c/d constant function.
Range = { a }
domain xR
2.
Polynomial function
: A function y = f(x) = a0xn+ a1xn-+ ...... + an
where a0, a1, ....... an
are real constants & n is non -ve integer, then f(x) is c/d polynomial function. If a0= 0, then n is degree of polynomial function.
Graph of f(x) = x2
f(x) = x2
is called square function. DomainRRangeR+  {0} or [0,]
Graph of f(x) = x3
:
f(x) = x2is cube functiondomainRRangeR
(3)
Rational Function
: It is ratio of two polynomialsLet P(x) = a0xn+ a1xn - 1+ .......... + an
Q(x) = b0xm+ b1xm - 1+ .......... + bm Then f(x) =
is a rational function if Q(x)0DomainR{x | Q(x) = 0}i.e. Domain
R except those points for which denomiator = 0
Graph of f(x) =  f(x) =
is called reciprocal function with coordinate axis as asymptotes.
DomainR - {0}RangeR - {0}
Graph of f(x) = 
f(x) =
DomainR - {0}Range(0,)
Dumb Question
: For y =
, how domain & range is R - {0} ?
Ans: For domains f(x) = + &  f(x) = -
So, f(x) is not defined at x = 0
Similarly for Rangeas x± f(x)0But we exclude ± So, RangeR - {0}(4)
Irrational Function
: Algebraic function containing trems having non-integral rational powers of x are c/d irrational functions.
Graph of f(x) =  f(x) = DomainR+ {0} or {0,)RangeR+ {0} of [0,)
Grafh of f(x) =  f(x) =
domainsRRangeR
(5)
Identity Function
: The function y = f(x) = x for all x
R c/d identity function on R
DomainRRangeR
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Vriti Edustore
