functions-and-graphs-1
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 : A B
& 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
x2 element in set A take n images
.............................................
.............................................
xm can 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, 4th root)
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: 10x + 10y = 10 10y = 10 - 10x
y = log10(10 - 10x) {am = b m = logab}
Now, 10 - 10n > 0
101 > 10x 1 > x
domain 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 domain R 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 x
ln is not defind,
D = 16 - 4 x 3 x 5 = - 44 < 0
& coeff. of x2 = 3 > 0
3x2 - 4x + 5 > 0 v x R
domain R
y = loge(3x2 - 4x + 5) 3x2 - 4x + 5 = ey
3x2 - 4x + (5 - ey) = 0
Since x is real, So, D 0
(- 4)2 - 4(3)(5 - ey) > 0 ey > 
y log
range [log, )
1. Constant Function: If range of function f consists of only one no. then f is c/d constant function.
Range = { a }
domain x R
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. Domain R
Range R+ 
{0} or [0, ]
Graph of f(x) = x3:
f(x) = x2 is cube function
domain R
Range R
(3) Rational Function: It is ratio of two polynomials
Let P(x) = a0xn + a1xn - 1 + .......... + an
Q(x) = b0xm + b1xm - 1 + .......... + bm
Then f(x) = is a rational function if Q(x) 0
Domain R {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.
Domain R - {0}
Range R - {0}
Graph of f(x) =
f(x) =
Domain R - {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 Range
as x ± f(x) 0
But we exclude ±
So, Range R - {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) =
Domain R+ {0} or {0, )
Range R+ {0} of [0, )
Grafh of f(x) =
f(x) =
domains R
Range R
(5) Identity Function: The function y = f(x) = x for all x R c/d identity function on R
Domain R
Range R
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 : A
B& 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
x2 element in set A take n images
.............................................
.............................................
xm can take n images.
Total no. of function from A to B n x n x ............... m times = nmDomain: 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, 4th root)
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: 10x + 10y = 10
10y = 10 - 10x y = log10(10 - 10x) {am = b m = logab} Now, 10 - 10n > 0
101 > 10x 1 > x domain 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 domain
R 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 > 0
3x2 - 4x + 5 > 0 R domain R y = loge(3x2 - 4x + 5)
3x2 - 4x + 5 = ey 3x2 - 4x + (5 - ey) = 0 Since x is real, So, D
0 (- 4)2 - 4(3)(5 - ey) > 0
ey > y log 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 x
R 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. Domain
R Range
R+ {0} or [0, ] Graph of f(x) = x3:
f(x) = x2 is cube function
domain
R Range
R (3) Rational Function: It is ratio of two polynomials
Let P(x) = a0xn + a1xn - 1 + .......... + an
Q(x) = b0xm + b1xm - 1 + .......... + bm
Then f(x) =
is a rational function if Q(x) 0 Domain
R {x | Q(x) = 0} i.e. Domain
R except those points for which denomiator = 0Graph of f(x) =
f(x) =
is called reciprocal function with coordinate axis as asymptotes. Domain
R - {0} Range
R - {0} Graph of f(x) =
f(x) =
Domain
R - {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 Range
as x
± f(x) 0 But we exclude ±
So, Range
R - {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) =
Domain R+ {0} or {0, )Range
R+ {0} of [0, )Grafh of f(x) =
f(x) =
domains
R Range
R (5) Identity Function: The function y = f(x) = x for all x
R c/d identity function on R Domain
R Range
R