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, .......... , x_m}      F : A B
&   B ={y₁, y₂, y₃, .......... , 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


Dumb Question: How total no. of function from A to B = n^m ?

Ans: x₁, element in set A take n images
       x₂ 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: 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, 4^th 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. 10^x + 10^y = 10.

Ans: 10^x + 10^y = 10   10^y = 10 - 10^x
y = log₁₀(10 - 10^x)      {a^m = b   m = log_ab}
Now,    10 - 10ⁿ > 0
10¹ > 10^x   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 = log_e(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 = log_e(3x² - 4x + 5) 3x² - 4x + 5 = e^y
3x² - 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, )




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 + a₁x^n- + ...... + a_n where a₀, a₁, ....... a_n 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 + a₁x^{n - 1} + .......... + a_n
       Q(x) = b₀x^m + b₁x^{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, )




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