Function:

A function y = f(x) is a rule which assigns a unique value of “y” to each value of “x” e.g   y = x^2

Notations:

y = x^2

f(x) = x^2

f : x = x^2

Evaluation of a function at a certain value of independent variable is given by

f(x) = 2x^2 + 3

f(1)= 2(1)^2 + 3

=2 + 3

=5 (function value)

Note: 

Functions are major tools to describe real world problems into mathematical forms e.g

Displacement ‘S’ and Velocity ‘V’ of a moving object is a function of time ‘t’:

S = 3t^2 + 2t + 1

V = 6t + 2

Domain:

                   Set of those values of “x”  for which a function is defined and real valued is called domain

Example:

(1)  y = x^2

Domain = R or (positive infinity to negative infinity)

(2) y = 1/x

Domain = (- infinity , 0) U (o , + infinity)

(3) y = 1 / x-1

Domain = {1}

(4) y = sqrt(1 – x^2)

1 – x^2 >= 0

-x^2 >= -1

x^2 <= 1

sqrt(x) <= sqrt(1)

we know that sqrt(x) is equal to |x| so inserting in above equation

|x| <= 1

-1 <= x <= 1

Domain = [-1 , 1]

Range:

                 Set of values of “y” corresponding to values of “x” in domain is called range

Example

(1) y = x^2

Range = All non negative real numbers = [0 , +infinity)

Even Function:

A function ‘‘ is called even function if f(-x) = f(x) e.g

f(x) = cos(x)

f(-x) = cos(-x) = cos(x) = f(x)

Odd Function:

A function ‘‘ is called odd function if f(-x) = -f(x) e.g

f(x)=sin(x)

f(-x) = sin(-x) = -sin(x) = -f(x)

Greatest Integer Function (Integer Floor Function):

[x] = Greatest integer not greater than “x”  e.g

(1) [0] = 0

(2) [0.5] = 0

(3) [3.2] = 3

(4) [-0.5] = -1

Least Integer Function (Integer Ceiling Function)

[x] = Smallest integer not smaller than “x”  e.g

(1) [0.5] = 1

(2) [2] = 2

(3) [3.3] = 4

Note:  x-1 <= [x] <= x

Piecewise Defined Function

Modulus Function

Unit Step Function

Composite Functions:

If “” and “g” are functions then composite function fog is defined as fog(x) = f(g(x))

e.g

f(x) = x^2   ,   g(x) = 2x + 1

fog(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2

gof(x) = g(f(x)) = g(x^2) = 2x^2 + 1

fof(x) = f(f(x)) = (x^2)^2 = x^4

gog(x) = g(g(x)) = 2(2x + 1) + 1 = 4x + 3

Hyperbolic Functions:

sinhx = e^x – e^-x / 2

coshx = e^x + e^-x / 2

tanhx = e^x – e^-x / e^x + e^-x

cothx = e^x + e^-x / e^x – e^-x

cosechx = 2 / e^x – e^-x

sexhx = 2 / e^x + e^-x

Graphs of Functions:

 

Piecewise function of above graph is f(x)={1    if x>=0

0     if x<=0}

Red line shows direct current which is 1 for all x values and Blue line shows alternating current which is obtained by sin(x).

Shifting of Graphs:

There are two types of shifting

Vertical Shift

The graph of function y = f(x) will shift “h” units above if it’s definition changes to y = f(x) + h

Horizontal Shift

The graph of a function y = f(x) will shift “h” units left if it’s definition changes to y = f(x+h)

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