# 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

__: What is non empty set ?__

**Dumb Question**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 = {x

_{1}, x

_{2}, x

_{3}, .......... , x

_{m}} F : A B

& B ={y

_{1}, y

_{2}, y

_{3}, .......... , y

_{n}}

Then if each elemenet in set A has n images in set B.

Thus, total no. of functions from A to B = n

^{m}

__: How total no. of function from A to B = n__

**Dumb Question**^{m}?

Ans: x

_{1}, element in set A take n images

x

_{2}element in set A take n images

.............................................

.............................................

x

_{m}can take n images.

Total no. of function from A to B

n x n x ............... m times = n

^{m}

__: Domain of y = f(x) is set all real x for which f(x) is defined (real).__

**Domain**__:__

**How to find Domain**(i) Expression under even root (i.e. square root, 4

^{th}root) 0

(ii) Denominator 0

(iii) If domain of y = (x) & y = y(x) are D

_{1}& D

_{2}respectivly then domain of f(x) ± y(x) of f(x).g(x) is D

_{1}D

_{2}.

(iv) Domain of is D

_{1}D

_{2}- {g(x) = 0}.

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

**Illustration**^{x}+ 10

^{y}= 10.

Ans: 10

^{x}+ 10

^{y}= 10 10

^{y}= 10 - 10

^{x}

y = log

_{10}(10 - 10

^{x}) {a

^{m}= b m = log

_{a}b}

Now, 10 - 10

^{n}> 0

10

^{1}> 10

^{x}1 > x

domain x (- , 1)

__: What is single value Function ?__

**Dumb Question**Ans. For every point of domains, these is unique image only.

__: Range of y = f(x) is collection of all output & {F(x)} corresponding to each real no. is domain.__

**Range**__:__

**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.

__: Find range of function y = log__

**Illustration**_{e}(3x

^{2}- 4x + 5).

Ans: y is difind if 3x

^{2}- 4x + 5 > 0

[

__Dumb Question__: Why 3x

^{2}- 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 x

^{2}= 3 > 0

3x

^{2}- 4x + 5 > 0

domain R

y = log

_{e}(3x

^{2}- 4x + 5) 3x

^{2}- 4x + 5 = e

^{y}

3x

^{2}- 4x + (5 - e

^{y}) = 0

Since x is real, So, D 0

(- 4)

^{2}- 4(3)(5 - e

^{y}) > 0 e

^{y}>

y log

range [log, )

**CLASSIFICATION OF FUNCTIONS**1.

__: If range of function f consists of only one no. then f is c/d constant function.__

**Constant Function**Range = { a }

domain x R

2.

__: A function y = f(x) = a__

**Polynomial function**_{0}x

^{n}+ a

_{1}x

^{n-}+ ...... + a

_{n}where a

_{0}, a

_{1}, ....... a

_{n}are real constants & n is non -ve integer, then f(x) is c/d polynomial function. If a

_{0}= 0, then n is degree of polynomial function.

**Graph of f(x) = x**^{2}f(x) = x

^{2}is called square function. Domain R

Range R

^{+}{0} or [0, ]

__:__

**Graph of f(x) = x**^{3}f(x) = x

^{2}is cube function

domain R

Range R

(3)

__: It is ratio of two polynomials__

**Rational Function**Let P(x) = a

_{0}x

^{n}+ a

_{1}x

^{n - 1}+ .......... + a

_{n}

Q(x) = b

_{0}x

^{m}+ b

_{1}x

^{m - 1}+ .......... + b

_{m}

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, )

__: For y = , how domain & range is R - {0} ?__

**Dumb Question**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)

__: Algebraic function containing trems having non-integral rational powers of x are c/d irrational functions.__

**Irrational Function**

**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)

__: The function y = f(x) = x for all x R c/d identity function on R__

**Identity Function**Domain R

Range R