#### 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 ‘*f *‘ 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 ‘*f *‘ 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 “*f *” 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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