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Functions and Graphs
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Introduction to Functions

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) Denominator0

(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 ofis 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 domainsf(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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