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

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

If 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 ={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.


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)

(ii) Denominator

(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

(iv) Domain of


Illustration: Find domain of single valued function y = f(x) given by eq. 10x + 10y = 10.
Ans: 10x + 10y = 10



Now, 10 - 10n > 0


domain x


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



(ii) If domain

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

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 = -

ln is not defind,
D = 16 - 4 x 3 x 5 = - 44 < 0
& coeff. of x2 = 3 > 0




y = loge(3x2 - 4x + 5)


Since x is real, So, D

(- 4)2 - 4(3)(5 - ey) > 0









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


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

Range




Graph of f(x) = x3:
f(x) = x2 is cube function
domain

Range


(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) =


Domain


i.e. Domain

Graph of f(x) =

f(x) =

Domain

Range


Graph of f(x) =

f(x) =

Domain

Range



Dumb Question: For y =

Ans: For domains




So, f(x) is not defined at x = 0
Similarly for Range
as x




But we exclude ±

So, Range

(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) =




Range




Grafh of f(x) =

f(x) =

domains

Range


(5) Identity Function: The function y = f(x) = x for all x

Domain

Range

