The explicit function is a function in which the dependent variable has been given “explicitly” in terms of the independent variable. Or it is a function in which the dependent variable is expressed in terms of some independent variables.

It is denoted by:

y=f(x)

Examples of Explicit functions are:

y=axⁿ+bx where a , n and b are constant.

y=5x³-3

The Implicit function is a function in which the dependent variable has not been given “explicitly” in terms of the independent variable. Or it is a function in which the dependent variable is not expressed in terms of some independent variables.

It is denoted by:

R(x,y) = 0

Some examples of Implicit Functions are:
x² + y² – 1 = 0

y⁴ + x³ +17 = 0

Although you can convert a Implicit function into Explicit function it is generally not done because after conversion
the new explicit function becomes very complex and some times also gives two different function branch.
For example you can convert the first implicit function example above to a Explicit function but after conversion it gives the following function:



And in the new function there are two branches of “y” one the positive branch and another the negative branch.

Establishing some basic properties or relations of Logarithms helps

us for faster computation of mathematical problems.

Some of the basic properties of Logarithms and proof of them are given below, they are also called theorems of logarithm.

1>  For any positive x ,y and a:

Proof:

Suppose:

then, and

so,

thus

2> For any positive a and x:

Proof:

suppose,

then

and

thus, 

3> For any positive x , y and a:

proof:

4> For any positive a,b and x:

Proof:

suppose ,

then ,

thus ,

If “x” is a number and,

Then , “y” is known as the logarithm of “x” to the base “a”.

For example:
:-The logarithm of 16 to the base 2 is 4 which can be shown as:

:-The logarithm of 8 to the base 2 is 3 which can be shown as:

The logarithmic Function:

The function in which the relationbetween input(let x) and and output(let y)

is given by:

 , where “a” is a constant,

is known as the logarithmic function.

when,

,

“y” is known as the logarithmic function of “x”  to the base “a”. and denoted as:

Relation between Exponential function and Logarithmic function:

Exponential function and Logarithmic are inverse function of each other if they are on the same base.

For example:

:- When ,

“8″ is the  Exponential function of “3″ and “3″ is the logarithmic function of “8″ to the base “2″.

Note:

:- Logarithmic function to the base 10 is known as common logarithm.

:- Logarithmic function to the base “e” is known as natural function where the value of “e” is given by:

and if the base of a logarithmic function is “e” then the base is usually omitted and written as:

=

Exponential function is the function which is defined by the following formula:

y=f(x)=a^x

Where , a is a constant great than 0 and a , x both are real numbers.

In any exponential function defined by formula y=f(x)=a^x,

“a” is said to be the base of exponential function “f” and “x” is the exponent of “a”.

Some Examples of Exponential Function:

a> if f:A→B is defined by f(x)=2^x then “f” is exponential function of base 2. We can show this exponential function in graph as:

b. If g:A→B is a function defined by f(x)=1/2^x then “f” is exponential function of base “1/2″ we can show this exponential function in graph as:

When a number (let a) is multiplied by itself multiple time(let n times) then we represent this case by adding “n” superscription to “a” as given below:

aⁿ

When , a number is repeatedly multiplied by itself the number of times the number is multiplied by itself is called the exponent the the number which is being multiplied is called base or index.

For example:

In aⁿ

“a” the index or base and “n” is exponent of the base “a”.

Laws of Exponents or Indices:

The rules which gives meaning to expression with exponent is known as laws of indices. And the laws of indices are:

1. aⁿ = a×a×a×…………..n^th Term

2. a^-n=1/aⁿ

3.

4.  a^p / a^q = a^p-q

5. a^m×aⁿ=a^m+n

Note:

Many thanks to Co. H . Tran  for his help to make this page better.

If we place an angle in standard position or at origin and draw a circle with center at origin such that the circle will intersect terminal arm of the angle , As shown in following figure:

Where “x” is the x-coordinate of the point “P”, “y” is the y-coordinate of point “P” and “r” the radius of circle.

Then for any angle of Θ there are six trigonometric functions named: Sine , Cosine , Tangent , Cosecant , Secant and Cotangent

The above functions are defined by :

Sine of the angle Θ = SinΘ=y/r

Cosine of the angle Θ=CosΘ=x/r

Tangent of the angle Θ=TanΘ=y/x

Cosecant of the angle Θ=CosecΘ=r/y

Secant of the angle Θ=SecΘ=r/x

Cotangent of the angle Θ=CotΘ=x/y

Note:

:-Cosecant , Secant and Cotangent are just inverse of Since , Cosine and Tangent respectively.

:-For some of the cases (when the denominator of the value of function if 0) or values of Θ Cosecant , Secant , and Cotangent may not be defined.

IF a line is rotated on one of it’s end point without changing the position of it’s another end point or it is rotated on one of it’s end point the the configuration formed by initial arm , terminal arm and vertex or common end point is known as Angle.

An angle always has following things:

An initial arm.

A terminal arm.  and

A vertex.

As shown in figure below:

An angle is said to be positive if the initial arm is rotated anti-clockwise direction and vice versa.

If the vertex of any angle is at the origin of a rectangular co-ordinate system the it is said to be in the standard position.

Often , in problem related to angle we assume the angle to be in standard position.

Measurement of Angle:

An angle is measured in terms of the amount of rotation done by initial arm in an angle. To measure an angle we assume an angle as standard or a unit of angle and compare the unknown angle with it.

On the basis of the definition of standard angle there are many types of angle measurement system the most commonly used two are described below:

Sexagesimal System.

In this system. The unit of angle measurement is a degree.

If aright angle(one fourth of a complete rotation) Is divided into 90 equal parts then each part or angle is regarded as a degree of angle. For smaller angle measurement a degree is divided into 60 equal parts and each part or angle is regarded to be a minute of angle. For further smaller angle measurement a minute is further divided into 60 equal parts and each part or angle is regarded as a second of angle.

A degree is denoted by “°” , minute by “′” and a second by “″”

The angle of measurement in this system can be summarized as:

60″(60 seconds)=1′(1 minute)

60′(60 minute)=1°(1 degree)

90°(90 degree)=1 right angle.

Radian Measure:

In this system the unit of angle measurement is  a “radian”.

A radian is an angle subtended at the center of a circle by an arc equal in length to the radius of circle.

For example:

The angle θ is a radian if the arc AA′ is equal to the radius of circle in given figure:

A radian is denoted by the symbol “^c”

Note: In any circle circumference = 2πr

so, 360°=2π^c

Algebraic functions are also known as Polynomial functions.

a. The Identity Function:

The function defined by y=f(x)=x is known as identity function.

For example:

If I:R→R is a  function defined by f(x)=x=y then the function “I” is a identity function.

We can show a identity function in graph as:

b. The Constant Function:

The function defined by y=f(x)=k , where “k” is a constant is known as constant function.

For example:

If A={1,2,3} and b={1} and f:A→B defined by y=f(x)=1 then “f” is a constant function.

We can show a constant function in graph as:

c. The Linear Function:

The function defined by y=f(x)=mx+c where “m” and “c” are constants is known as linear function.

For example:

If f:A→B is defined by y=f(x)=2x+1 then “f” is a linear function , where m=2 and c=1

We can show linear function in graph as:

d. The quadratic function:

The function defined by y=f(x)=ax²+bx+c is known as quadratic function where “a” , “b” and “c” are constants.

For example:

If f:A→B is defined by y=f(x)=x² then “f” is a quadratic function where a=1 and b=c=0

We can show quadratic function in graph as:

e. The cubic Function:

The function defined by y=f(x)=ax³+bx²+cx+d is known as cubic function.

For example:

If f:A→B is defined by y=f(x)=x³ then “f” is a cubic function where a=1 and b=c=d=0

We can show cubic function in graph as:

Let f:A→B and g:B→C be any two functions , we can represent this two functions in a diagram as following.

What is happening here is the output of function “f” is again processed by another function “g” to give a new output.

If we make a new function (let the new function be “h”) which maps input of the function “f” directly to output of function “g” or maps “A” directly to “C” then the new function “h” is said to be the composite function of functions “f” and “g”.

Thus,

A single function formed by the combination of two or more function is said to be a composite function.

NOTE:

Composite function of “f” and “g” is not same as the composite function of “g” and “h”.

Denotation of Composite Function:

We denote a composite function of two functions (let “f” and “g”) by adding a “o” sign in between them , and the two functions “f” and “g” are ordered on the basis of which occurs first.

Like we denote composite function of f:A→B and g:B→C as:

gof

NOTE:The function which occurs at first is placed in second place of composite function symbol and vice versa.

The composite function of  Functions “f” and “g” in above diagram or “gof” is shown diagrammatically as following:

Some Examples of Composite Function.

a> If f:A→B and g:B→C are two functions defined by:

Then , (gof):A→C or composite function of “f” and “g” is defined by:

b> If f:R→R is defiend by f(x)=3x and g:R→R is defined by g(x)=x+1 and “R” is the set of real numbers.

Then ,

(gof) is defined by:

(gof)(x)=g(f(x))

=g(3x)

=3x+1

In above calculation , in g(f(x)) first function “f” acts on “x” and changes it to “3x” as “f’ is defined and changes g(f(x)) into g(3x) and then , function “g” acts on “3x” to change it to “3x+1″ is function “g” is defined.

Let “f” be a Function from set A to B or , elements of set A are changed or outputted to elements of set B if they are processed through or inputed in the function “f”.

or f:A→B

Then , inverse of the function “f” is a new  rule or function (Let the new function be function “g”)  , such that if we input the output of function “f” into the new function “g” then it produces or outputs the input of function “f”

or,

inverse of a function “f”  f:A→B is another function”g” such that g:B→A

Denotation of Inverse Function:

As we know , A function(f) from set A to set B is denoted by:

f:Ar→B

the inverse of the function “f” is denoted by adding a “-1″ superscription in the function and interchanging the position of input and output set of the function. Like inverse of function “f” denoted above is denoted as:

f^-1:B→A

We can denote the inverse of a function diagrammatically as following:

And , as we denoted the process of processing of input (x) by a function (f) and producing output (y) by:

f(x)=y

We denote the same fact for inverse of the function (f) by:

f^-1(y)=x

which means the inverse function “f^-1” processes “y” which is output of function “f” and outputs “x” which is input of function “f”.

Note:

f^-1≠1/f

There is no Inverse of Onto Function:

A function “f’ can only have a defined inverse function “f^-1“only if it is a One-To-One function. A Onto Function can never have a inverse practically.

For Example:

If A={1,2,3} and B={a,b} and function “f” is defined by:

Then , we cannot make a inverse of function “f” because inverse of function “f” should map “a” to “1″ and “b” to “2″ and “3″ both.

But practically we can never make a rule or function which have two or variable output of same input.

Some Examples of Inverse Function:

a> Let f:A→B be a function which is defined by the arrow diagram on the top of the image below. Then the inverse function of “f” or , f^-1:B→A is defined by the arrow diagram on the bottom of the image below.

b> If f:A→B is a function defined by f(x)=x² then , the inverse function of “f” is “f^-1” which is defined by:
f(x²)=x
or, f(x)=√x