Functions:

Functions:

Introduction to Functions in Math:

The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type.(source : Wikipedia)

In this article we are going to see about functions in math,types of functions in math and some sample solved problems on functions in math.

Types of Functions in Math:

There are several types of functions in math.the following are the some of the different types of functions,

*

Composite functions
*

Inverse functions
*

Trigonometric functions
*

Quadratic functions
*

Linear functions
*

Cubic functions
*

Algebraic functions
*

Periodic functions
*

Even and odd functions

Next we are going to see some problems on functions in math.

Solved Problems in Functions in Math:

Problem 1:

Function f(x) =5x2 -3 and g(x) = 6x. Find f(g(x)).

Solution:

Given, f(x) =5x2 -3 and g(x) = 6x .

To find the f(g(x)) substitute x = g(x) in f(x),

f(x) =5x2 -3

Substitute x = g(x) = 6x

f(g(x)) = 5(6x)2 -3

= 5 ( 36x2 ) -3

= 180x2 - 3

Answer: f(g(x)) = 180x2 - 3

Problem 2:

Find the following function is odd or even f(x) = 10x2 + 5

Solution:

Given f(x) = 10x2 + 5

We need to find the function is odd or even.

If f(-x) = f(x) , the given function is even

If f(-x) = - f(x) , the given function is odd

So to find the function is odd or even ,Substitute x = -x in the given function,

f(-x) = 10(-x2 ) + 5

= 10x2 + 5

= f(x)

f(-x) = f(x)

So the given function is even function.

Answer: Given function f(x) =10x2 + 5 is even function.




Problem 3:

Find the inverse function of f(x) = 15 x - 8

Solution:

Given, f(x) = 15 x - 8

Let us take f(x) = y

That is y = 15 x - 8

To find the inverse function we need to solve for x,

y = 15 x - 8

Add 8 on both sides,

y + 8 = 15x - 8+ 8

y + 8 = 15x

15x = y + 8

Divide by 15 on both sides,

x = ( y +8) / 15

Now replace y = x and x =f--1 ( x)

f--1( x) = (x+8)/15

Answer: Inverse function of a given function is f--1( x) = (x+8)/15

Hope you like the above example of Functions in Math.Please leave your comments, if you have any doubts.

Sets

When we usually hear about the word Set,the first thing that would come to our mind would be a collection of something.In Mathematics, the basic meaning of set is a "well-defined collection of definite objects is called a set."

George Cantor is regarded as the "Father of Set theory".

The concept of "Sets" is basic in all branches of mathematics.

Set: Definition: well-defined collection of distinct objects is called a set.Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

Notation of Sets: Capital letters are usually used to denote or represent a set.

Representation of Sets: There are two methods of representing a set. (i) Roster Method (ii) Set builder form.

Finite and Infinite Sets: A set is finite if it contains a specific number of elements. Otherwise, a set is an infinite set.

Null Set or Empty Set or Void Set: A set with no elements is an empty set.

Singleton Set or Singlets: A set consisting of a single element is called a singleton set or singlet. The cardinality of the singleton set is 1.

Equivalent Sets: Two finite sets A and B are said to be equivalent sets if cardinality of both sets are equal i.e. n (A) = n (B).

Equal Sets: Two sets A and B are said to be equal if and only if they contain the same elements i.e. if every element of A is in B and every element of B is in A. We denote the equality by A = B.

Cardinality of a Set A: The number of elements in a finite set A, is the cardinality of A and is denoted by n(A).

Universal Set: In any application of the theory of sets, the members of all sets under consideration usually belong to some fixed large set called the universal set.

Subsets: If A and B are sets such that each element of A is an element of B, then we say that A is a subset of B and write A Í B.

Power Set: The family of all subsets of any set S is called the power set of S. We denote the power set of S by P (S).

Hope you like the above example of Sets.Please leave your comments, if you have any doubts.

Algebraic Expression:

Introduction:In algebra an expression may be used to designate a value, which value might depend on values assigned to variables occurring in the expression; the determination of this value depends on the semantics attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance an expression might designate a condition

Algebraic expression comes under basic algebra category in which algebra is one of the main branches in mathematics which deals with finding the unknown variable value with the references of known values. In algebraic expression the variables are represented with the help of alphabetic letters and the integers used in the algebraic expressions are considered as constants. The algebraic expressions can be easily solved by using arithmetic operations. The following are some of the example problems in algebraic expressions.

Algebraic Expressions Example Problems:

Example 1:

Derive the algebraic expression.

-2(m - 1) - 4m - 1 = 3(m + 2) - 2m

Solution:

Given
-2(m - 1) - 4m - 1 = 3(m + 2) - 2m

Multiply factors
-2m + 2 - 4m - 1 = 3m + 6 - 2m

Group like terms
-6m + 1 = m + 6

Subtract 1 to both sides
-6m + 1 - 1 = m + 6 -1

Group like terms
-6m = m + 5

Subtract m to both sides
-6m - m = m + 5 -m

Group like terms
-7m = 5

Multiply both sides by -1/7
m = - 5/7

m = - 5/7 is the solution to the given equation

Example 2:

Derive the algebraic expression.

-4(m + 2) = m + 9

Solution:

Given
-4(m + 2) = m + 9

Multiply factors in left term
-4m - 8 = m + 9

Add 8 to both sides
-4m - 8 + 8 = m + 9 + 8

Grouping the above terms
-4m = m + 17

Subtract m to both sides
-4m - m = m + 17 -m

Group like terms
-5m = 17

Multiply both sides by -1/5
m = -17/5

m = -17/5 is the solution to the given equation.

Hope you like the above example of
Algebraic Expression.Please leave your comments, if you have any doubts.

Simplification Example Problems

Simplification Example Problems:

Meaning of Simplification:


Simplification calculator is the online software in the Internet through which students in online give the question for simplification they get the answer to their questions. These simplification calculator is very much helpful for students to clear their doubts in their home work problems. Sample problems from algebra involves step by step simplification. The simplification calculator for algebra would be much useful for school students. let us see some sample problems from algebra for simplification calculator.

Example 1:


Find the sum of 2x4 – 3x2 + 5x + 3 and x + x3 – 6x2 – 1.

Solution:

Using the associative and distributive property, we obtain

(2x4 – 3x2 + 5x + 3) + (x3 – 6x2 + x – 1) = 2x4 + x3 – 3x2 – 6x2 + 5x + x + 3 – 1

= 2x4 + x3 – (3+6)x2 + (5+1)x + 2

= 2x4 + x3 – 9x2 + 6x + 2. (answer)

Example 2:

Factorize x2 – 8xy – x + 2y.

Solution:

x2 – 8xy – x + 2y = (x2 – 8xy) – (x – 2y)

= x(x – 8y) + (–1) (x – 2y)

= (x – 8y) [x + (–1)]

= (x – 8y) (x – 1). (answer).






Example 3:

Problem to find the variables x,y and z:

x + y + 2z = 8 ------> (1)

3x - y + 3z = 4 ------>(2)

2x + y + 4z = 6 ------>(3)

Solution:

Solve (1) and (2),

x + y + 2z = 8 ----> (1)

3x - y + 3z = 4 -----> (2)

add the above two equations.

we get 4x + 5z = 12 ------> (4)

solve (1) and (3)

(3) * 2 ----> 4x + 4y +8z = 16

(4) -----> 4x + 0y +5z = 12
(-) (-) (-)

3z = 4

[ z = 4/3]

substitute [z = 4/3] in (4) eqn

4x + 5 [(4/3)] = 12

4x = 12 - [20/3]

4x = [(36-20)/3] = [16 / 3]

x = [16/12] = [ 4/3]

substitute x = [4/3] , z = [4/3] in (1) eqn.

[4/3] + y + 2 [(4/3)] = 8

y + [12/3 ] = 8

y + 4= 8

y = 8 - 4 .

x = [4/3] , y = 4, z = [4/3] .

Example 4:

Find the sum of x3y + 2x2y2 – 3xy3 and 3x3 – 3x3y + y3 + 4xy3.

Solution:

(x3y + 2x2y2 – 3xy3) + (3x3 – 3x3y + y3 + 4xy3) = x3y + 2x2y2 – 3xy3 + 3x3 – 3x3y + y3 + 4xy3

= (x3y – 3x3y) + (2x2y2) + (–3xy3 + 4xy3) + (3x3) + (y3)

= –2x3y + 2x2y2 + xy3 + x3 + 3x3 + y3

Hope you like the above example of Simplification.Please leave your comments, if you have any doubts.

Linear transformations:

Linear transformations:

Introduction:

Linear transformation is the most important topic in the algebra chapters in mathematics. Linear transformation in mathematics is a rule for changing one figure shape (or matrix or vector) into another by using a formula with a specified format. This format must be a linear combination, in which the original components are modified through the formula of ax + by to form the coordinates of the transformed figure. Stretching it or compressing it, and rotating it. All the transformation has an inverse.

Algebra Linear Transformation: In this linear transformation section we are going to take look at a special kind of function that the study of Linear Algebra and has many applications in fields of mathematics, physics and as well as the engineering field. This basic definitions and facts are combined with this kind of function. We are looking for more number of examples.We can understand about the Linear Transformation with the help of few examples. Here we have using two examples for Algebra Linear transformation,

Example1: Zero transformation:

Given: the zero transformation is the transformation T: Sn -> Sm that maps every vector in this format as y in Sn to the zero vectors in Sm, which is T(y) = 0.

Solution:

This is the zero matrix problem so let us take m*n is zero matrix,

And then the matrix induced by this linear transformation is the m*n is zero matrix, 0 matrix is since,

T(y) = T0 (y) = 0y = 0

There fore, to make it clear form of this matrix is, using the zero transformation for matrix, and we are usually denoted by this in T0 (y).

Example 2: Identity transformation:

Given: the identity transformation is the transformation T: Sn -> Sn ( both are Sn because of identity transformations) that maps every vector in this form as y itself to the identity. Which is T(y) = y.

Solution:

Then the matrix induced by this transformation is the n*n matrix (this is identity matrix), so that I n since,

T(y) = Ti (y) = In y = y

There fore, to make it formation of identity transformation is Ti (y) to make clear for this, and we are usually denoted by this in Ti (y).

Hope you like the above example of Linear Algebra.Please leave your comments, if you have any doubts.

Number Sense

Number Sense:

Introduction of Number Sense:Let me give an explanation about number sense in general,its very clear actually,as we can see the name of the concept itself is self explanatory,"Number Sense" number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations.Many other definitions exist, but are similar to the one given. Some definitions emphasize an ability to work outside of the traditionally taught algorithms, e.g., "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms.

Let us now study about the Definition of Number Sense:

Number sense is defined as the proper understanding and usage of the numbers in the various places

It includes various things

1. Ability to find the relative values of a number.

2. How to use a number in different arithmetic operations like (addition, subtraction, multiplication and division).

3. Finding the problem solving strategies.

Number sense covers various topics

They are

Estimating and rounding of the numbers

Rounding and addition

Rounding and product

Rounding and division


Hope you like the above example of Number Sense.Please leave your comments, if you have any doubts.

Law of Indices

Law of Indices:

Introduction:

Let us learn the meaning of Law of Indices in general,Indices is the concept of Algebra which is one of the major branch of mathematics.The Index of a number is the number of times multiplied by itself. If X and Y are positive integers and also X [!=] 0 then

X Y = X * X * X * X * X ........ Y times it is multiplied by same number

X is called the base and Y is the power. We read it as " X raised to the power Y " . The power is also called the " Index" or "Exponent".

A few examples of Powers : (i) 25 = 2 x 2 x 2 x 2 x 2 = 32 where index (or power or exponent) = 5 and base = 2 .

Its read as base 2 raised to the index 5 equals to 32

(ii) 23 = 2 x 2 x 2 = 8 where index = 3 and base = 2.

Its read as base 2 raised to the index 3 equals to 8

(iii) 33 = 3 x 3 x 3 = 27 where index = 3 and base = 3

Its read as base 3 raised to the power 5 equals to 27

(iv) 34 = 3x3x3x3 = 81 where index = 4 and base = 3

Its read as base 3 raised to the index 4 equals to 81

(v) 42 = 4x4 = 16 where index = 2 and base = 4

Its read as base 4 raised to the index 2 equals to 16

Indices or powers or Exponents are used to write statements involving repeated multiplication in shorthand.

Now let us learn about the specifics of the Law of Indices:For any expression like numbers , variables or functions having same base but different indices (or power or exponent) and also for the expressions consists of same index (or power or exponent) but different bases can be solved using Indices concept involving algebraic operations by following the below set of Laws of Indices.

Rule (1) Product Law for Indices : am x an = a m+n

consider an example, (a) 53 x 52 = 5 3+2 = 55 = 5 x 5 x 5 x 5 x 5 = 3125

(b) e 3t e -7t = e 3t x e -7t = e [3t + (-7t) ] = e [3t - 7t] = e -4t

(c) -4 4 -4 -1 = -4 [4 + (-1) ] = -4 [ 4 - 1] = -4 3

(d) (5/2) -3 (5/2) -9 = (5/2) -3 + (-9) = (5/2) -3 - 9 = (5/2) -12

(e) (q-2) (q 1/3) = q -2 + (1/3) = q (-6+1) / 3 = q -5/3

(f) (3) -1 (3) 5 = 3 -1+5 = 3 4

Rule (2) Quotient Law for Indices : am [-:] an =(a m / a n ) = a m-n

For instance, (a) 68 [-:] 65 = 6 8-5 = 6 3 = 6 x 6 x 6 = 216

(b) (4/3) -3 [-:] (4/3) 9 = (4/3) [-3 - 9] = (4/3) -12 = 1 / (4/3) 12 = (3/4) 12

(c) 5 -2 [-:] 5 -2 = 5 [-2 -(-2)] = 5 [-2 + 2] = 5 0 = 1

(d) g 6 [-:] g -7 = g [6 - (-7)] = g [6 + 7] = g 13

(e) Y (-7/2) [-:] Y (-1) = Y [ (-7/2) - (-1)] = Y [ (-7/2 + 1)] = Y [(-7+2)/2] = Y -5/2

Rule (3) Power Law for Indices : (a m) n = a mn

Example (a) (7 4) 2 = 7 4x2 = 7 8

(b) (2 -5 )4 = 2 (-5x4) = 2 -20

(c) (X -5) -3 = X (-5 x -3) = X -15

(d) [t (1/2)] 3 = t [(1/2)x3] = t (3/2)

Hope you like the above example of Law of Indices.Please leave your comments, if you have any doubts.

Congruent Triangles

Congruent Triangles:




Congruent Triangles:

The concept of Congruent Triangles is explained in detail here. Every geometric figure has a shape and a size. Two circles of different radii have the same shape but their sizes are different. But if we draw two circles of the same radius both the shape and size will be the same. Such figures with the same shape and size are called congruent figures. We can check whether two figures are congruent or not by the method of superposition. We say that two triangles ABC and DEF are congruent if we can make one triangle fit exactly on the other.

In the above figure, the two triangles are congruent and we write . The sides which are equal in length are called the corresponding sides and the angles which are equal in measure are called the corresponding angles.

Conditions for Congruent Triangles:

When two triangles have (i) the same shape, and (ii) the same size, then they are said to be congruent triangles.

Symbol for congruency is
Draw with AB = 4 cm, AC = 5 cm, and .

Also with DE = 4 cm, DF = 5 cm and

Draw these triangles on a tracing paper. Now place fig(ii) on fig.(i), so that D is placed on A, DE along AB and F on the same side as C.
It is noticed that coincides with exactly fits on ABC.


We Can put the entire explanation of Congruent Triangles in a Nut shell.


SAS Axiom:

If two triangles have two sides of one equal to two sides of the other, each to each, and angles included by these sides are equal, then the triangles are congruent.
AAS Axiom:

If two triangles have two angles of one equal to two angles of the other, each to each and also one side of one equal to the corresponding side of the other, then the triangles are congruent.
SSS Axiom:

If two triangles have three sides of one equal to three sides of the other each to each, then the triangles are congruent.
RHS Axiom:

If two right triangles have their hypotenuses equal and one side of one equal to one side of the other, then the triangles are congruent.

Hope you like the above example of Congruent Triangles.Please leave your comments, if you have any doubts.

Rhombus

Rhombus:

Introduction:

Rhombus is quadrilaterals with both pairs of opposite sides are parallel and all the sides are having same length. Rhombus is also called as equilateral parallelogram.
The rhombus is also called as diamond and rhomb. A square is also a rhombus, because the square is a quadrilateral with all sides equal.
All the opposite sides of the rhombus is parallel and all the opposite angles are equal.
One of the most important thing to keep in mind is the properties of a Rhombus.

• Opposite angles of the rhombus having equal measures.
• The diagonals of the rhombus intersect each other at right angles.
• Rhombus having two diagonals that are connecting opposite pairs of vertices.
• Rhombus is Symmetric along the diagonals
• Every Rhombus is parallelogram
• All the parallelograms are not a Rhombus.While studying about the Rhombus we cannot ignore to study about the area and the perimeter of the Rhombus.

The area of the Rhombus is same as the area of the parallelogram .That is the area is the multiplication of base and height.
The formula for calculate the area is,
A=b * h
The base of the rhombus is the length of one of its sides and the height is the perpendicular distance between the opposite sides. The perimeter of the rhombus is the total sum of its all the sides length. Generally the perimeter is sum of the length of the all sides. In the case of rhombus, all the sides are having equal length. So we can say that the perimeter is 4s, where s is length of the sides.
Hope you like the above example of Rhombus.
Please leave your comments, if you have any doubts.

Quadrilaterals:

Quadrilaterals:

A Quadrilaterals is a polygon with four sides (or 'edges') and four Vertices or Corners,its basically a figure formed by joining four points in an order is called a quadrilateral.Look around you and you will find so many objects which are of the shape of a
quadrilateral - the floor, walls, ceiling, windows of your classroom, the blackboard.....its very simple to identify a quadrilateral.The word Quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").
We can learn about Quadrilaterals by studying the objects around us as mentioned in the above example.The other way to understand quadrilaterals is by understanding the properties of a quadrilateral.

Let us now learn about the properties of a quadrilateral:

The sum of the angles of a quadrilateral is 360º.The other thing we must keep in mind while learning quadrilateral is that quadrilateral are of different shapes and sizes.
Rhombus:

A Rhombus or rhomb is a quadrilateral whose four sided all have the same length.The rhombus is often called a Diamond

Rectangle:

A rectangle is any quadrilateral with four right angles

Parallelogram:

A parralellogram is a quadrilateral with two pairs of parallel sides. In Euclidean Geometry, the opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

Hope you like the above example of Quadrilaterals
Please leave your comments, if you have any doubts.