Function in Math

Introduction for function in math:

Before giving the explanation of the function in math, we have to know, what is function in math? The function is nothing but the result of the given argument. The argument is also know as element of the function from the given set. Function is associated with domain and co domain. The element exists in domain which is also exactly one element in co domain.I like to share this Implicit Function with you all through my article.


Pre-requisite for function in math:


Constant and Variable:

The value will not be changed during the mathematical process is called constant. The value will be changed during the mathematical process is called variable.

Interval:

The subset of real number is called interval

Neighborhood:

In a number line the neighborhood of a real number is defined as a open interval of very small length.

Independent / Dependent Variable:

A variable is an independent variable when it has any arbitrary value. A variable is said to be dependent when its value depends on other variables.

Cartesian product:

The Cartesian product of the two sets A and B is denoted by A x B and is denoted as

Let A={ a1,a2,a3} B={b1,b2}

A x B={ (a1,b1),(a1,b2)(a2,b1)(a2,b2)(a3,b1)(a3,b2)}


Explanation of function in math:


A function is a special type of relation. If no two ordered pairs have same first element and different second element, the relation is called function. If two ordered pairs have same first element and different second element, the relation is called not a function. If the element x in the set A is associated with element x in the set B is called image of the function. The set of images is called range of the function. If the range of function in not equal to the co domain, those functions are called mapping.

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Types of function in math:


1. Identity function:

A function from a set A to the same set A is said to be an identity

2. Inverse of a function:

To define the inverse of a function f i.e. f−1 (read as ‘f inverse’), the

function f must be one-to-one and onto.

3. Constant function:

If the range of a function is a singleton set, the function is called a

constant function.

4. Linear function:

If a function f : R → R is defined in the form f(x) = ax + b then the function

is called a linear function. Here a and b are constants.

5. Polynomial function:

If f : R→R is defined by f(x) = an xn + an − 1 xn − 1+ …+ a1x + a0, where

a0, a1,…, an are real numbers, an≠0 then f is a polynomial function of degree n.

Integers and Math

Introduction to integers and math:
In mathematics, integer is one interesting topics in number representation. Integer has a set of numbers in which there are two difefrent types of integers that are non negative integers and negative integers. Integers have complete entity or unit. No fractional part in integers. Integers perform different arithmetic operations such as addition, subtraction, multiplication and division. Let us solve some example problems in integers and math.

Example for integers:

1254, -5365, 0 etc,

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Example problems for integers and math:


Example 1:

Perform arithmetic operation addition for the given two integer numbers

875 + 432

Solution:

Given two integer numbers are

875 + 432

Both are two positive integers so, the result is also a positive numbers

Here we add 875 into 432, and then we get the result

875 + 432

1307

Solution to the given two integers is 1307.

Example 2:

Perform arithmetic operation subtraction for the given two integer numbers

1024 and 827

Solution:

Given two integer numbers are

1024 - 827

Here we subtract 1024 into 827, and then we get the result

1024 - 827

197

Solution to the given two integers is 197.

Example 3:

Perform arithmetic operation multiplication for the given two integer numbers

1524 × 1028

Solution:

Given two integer numbers are

1524 × 1028

Both are positive integers so, the result is also a positive numbers

Here we multiply 1524 into 1028, and then we get the result

1524 × 1028

1566672

Solution to the given two integers is 1566672.

Example 4:

Perform arithmetic operation division for the given two integer numbers

6000/ 150

Solution:

Given two integer numbers are

6000 / 150

Both are positive integers so, the result is also a positive numbers

Here we divide 6000 by 150, and then we get the result

6000/ 150

40

Solution to the given two integers is 40.

Example 5:

Perform arithmetic operation addition for the given two integer numbers

- 728 + 206

Solution:

Given two integer numbers are

- 728 + 206

- 728 is negative number and 206 is positive number

Here we add - 728 into 206, and then we get the result

- 728 + 206

- 522

Solution to the given two integers is - 522.

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Practice problems for integers and math:


Perform arithmetic operation for the given two integers and math are,

i). 3286 + 7025

ii). 5021 – 2856

iii). 481 × 200

iv). 540 / 20

v). – 258 + 625

vi) – 286 × 85

Solution:

i). 10311

ii). 2165

iii). 96200

iv). 27

v) 367

vi) – 24310

Word Problems Second Grade Math

Introduction to word problem:
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a recursively presented group G is the algorithmic problem of deciding whether two words represent the same element. Although it is common to speak of the word problem for the group G strictly speaking it is a presentation of the group that does or does not have solvable word problem. (Source: Wikipedia)

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Example of word problems second grade math


Math second grade word problem: 1

There are 109 boxes in the store. We bought 42 more boxes for the store. How many boxes are there in the store now?

Solution:

In 109 boxes in the store

And then we bought 42 boxes in the store

109 + 42 = 151

Therefore 151 boxes in the store now.

Math second grade word problem: 2

A fruit wholesaler had 128 grapes. He sold 52 grapes. How many grapes did he have left?

Solution:

Wholesaler 128 grapes

Sold 52 grapes

Left grapes =?

So, 128 – 52 = 76

He had totally 76 grapes left.

Math second grade word problem: 3

There are 11 pine apples in each box. How many are there in 8 boxes?

Solution:

In one box = 11 pine apples

In 8 boxes =?

11 * 8 = 88

There are 88 pine apples in 8 boxes.

Math second grade word problem: 4

Andrew bought a sack of 62 kg of flour. He packed the flour equally into 8 bags. How many kilograms of flour in the each bag?

Solution:

In one sack = 62 kg

Then 3 bags =?

64 ÷ 8 = 8

There were 8 kg of flour in each bag.

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Practice - word problems second grade math


1. There are 1238 erasers in the store. We bought 53 more erasers for the store. How many erasers are there in the store now?

[Answer: 1291 erasers in the store now]

2. A vegetables seller had 563 potatoes. He sold 154 potatoes. How many potatoes did he have left?

[Answer: He had 162 oranges left]

3. There are 5 pine apples in each box. How many are there in 9 boxes?

[Answer: 45 pine apples in 9 boxes]

4. Arnold bought a sack of 90 kg of flour. He packed the flour equally into 10 bags. How many kilograms of flour in the each bag?

[Answer: 10 kg of flour in each bag]

Matrices in Math

Introduction to matrices in math:

In order to locate the position of a particular element of a matrix. We have to specify  the number of the row and that of  the column in which the element occurs. An element occurring in the ith row and jth column of a matrix A will be called the (i,j)th element of A , to be denoted by aij .

In general ,an m x n matrix A may be written as ,

A     =     `[[a_11,a_12,...,a_1n],[a_21,a_22,...,a_2n],[...,...,...,...],[a_m1,a_m2,...,a_mn]]`

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Types of matrices:


Row matrix:

Row matrix is a matrix that contain only one row or a row vector.

Example :

(i) A = [aij]1 × 3 = [1 6 5] is a row matrix of order 1 × 3.

(ii) B = [bij]1 × 2 = [0  1] is a row matrix of order 1 × 2

(iii) C = [cij]1 × 1 = [1] is a row matrix of order 1 × 1

Column matrix:

A colum matrix is a matrix that contain  only one column  or a column vector.

Example :

1)[aij]mxn =`[[2],[4],[6]]` is a column matrices of order 3 x1

2)[aij]mxn =`[[0],[1]]` is a column matrices of order 2 x1

3)[aij]mxn =[45] is column matrices of order 1 x1

Square matrix:

A square  matrix is a matrix in which the elments in rows and columns are equal and a  matrix of order n × n is also known as a square matrix of order n.

Example :

1) `[[4,5],[5,4]]` is the square matces of order 2 x2

2) `[[1,2,3],[4,5,6],[7,8,9]]` is the square matrix of order 3 x3

3) `[[0,1,2,3],[2,1,4,5],[3,4,6,7],[0,6,7,8]]` is the square matrix of order 4 x4.

Diagonal matrix:

Diagonal matrix means  a  square matrix A = [aij]n × n is said to be a diagonal matrix if aij = 0 when i ? j. In a diagonal matrix all the entries except the entries along the main diagonal are zero.

Example :

1) `[[1,0,0],[0,2,0],[0,0,3]]` is the diagonal matrix of order 3 x3

2) `[[3,0],[0,7]]`   is the diagonal matrix of order 2 x2.

Zero matrix or null matrix or void matrix:

All the entries are zero in the given matrix then the matrix is said ti be a zero  matrix .

Example :

A = `[[0,0,0],[0,0,0],[0,0,0]]` is the null matrix  or zero matrix of order 3 x 3

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Operation on Matrices in math:


Basic operation involved in matrix:

Addition of two matrices in math
Subtraction of matrices in Math
Multiplication of matrices in math.
Let  us discuss about the  operation of matrices in math.

1)Addition of matrices.

Let us consider the two different matrices of order 2 x2

A=`[[3,5],[4,7]]` and B= `[[1,8],[4,7]]`

A+B = `[[3+1,5+8],[4+4,7+7]]`

A+B =`[[4,13],[8,14]]`

Subtraction of two matrices:

Let us consider the two different matrices of order 2 x2

X= `[[4,6],[3,5]]` and Y =`[[5,7],[3,4]]`

X-Y =`[[4-5,6-7],[3-3,5-4]]`

X-Y =`[[-1,-1],[0,1]]`

Multiplication of two matrices:

Let us consider the two different matrices of order 2 x2

P = `[[4,5],[3,2]]` and Q =`[[2,3],[2,1]]`

PQ =`[[4*2 +5*2,4*3+2*1],[3*2+2*2,3*3+2*1]]`

PQ =`[[8+10,12+2],[6+4,9+2]]`

PQ =`[[18,14],[10,11]]`

6th Grade Math Probability

Introduction to 6th grade math Probability:

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. (Source: Wikipedia)


Example problems for 6th grade math probability:


Example: 1

Two dice are rolled. A is the event that the sum of the numbers shown on the two dice is 5, and B is the event that at least one of the dice shows up a 3. Are the two events

(i) Mutually exclusive

(ii) Exhaustive? Give the arguments in support of your answer.

Solution:

When two dice are rolled, we have n(s) = (6 x 6) = 36.

Now, A = {(1, 4), (2,3),(4,1),(3,2)} and

B = {(3,1), (3,2), (3,3), (3,4), (3, 5), (3,6), (1,3), (2,3), (4,3), (5,3), (6,3)}

(i) A`nn`B = {(2,3), (3,2)} `!=` `phi` .

(ii) Also, A`uu`B `!=` s

Therefore A and B are not exhaustive events.

Example: 2

Two dices are rolled. Let A, B, C  be the event getting a sum of 2, a sum of 3 and a sum of 4 respectively. Then, show that

(i)    A is a simple event

(ii)   B and C are compound events

(iii)  A and B are mutually exclusive

Solution:

Clearly we have

A  = {(1, 1), B = (1,2),(2,1) and c = (1,3),(3,1),(2,2)}

(i) Since A consists of a single sample point, it is a simple event.

(ii) Since both B and C contain more than one sample point, each one of them is compound event.

(iv) Since A`nn` B =`phi`, A and B are mutually exclusive.

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Practice problems for 6th grade math probability:


1. From a well-shuffled deck of 52 cards, a card is drawn at random. Find the probability that the card drawn is

(i) red and a  king

(ii) Either red or a king

[Answer: (i) 1/26

(ii) 7/13]

2.The odds in favour of occurrence of an event are 5:12. Find the probability of the event.

[Answer: 5/17]

Answer Math Problems For You

Introduction to answer math problems:

Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. I like to share this Small Sample Size Statistics with you all through my article.

Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and problems of game theory.

SOURCE: WIKIPEDIA


Examples of answer math problems:


Math problem 1:

Solve the equation 5(-3y - 2) - (y - 3) = -4(4y + 5) + 13

Solution:

Given the equation 5(-3y - 2) - (y - 3) = -4(4y + 5) + 13

First we can multiply the factors for both left hand side and right hand side.

-15y - 10 - y + 3 = -16y - 20 +13

Here we can group the like terms on both sides. Then we get.

-16y - 7 = -16y – 7

Add 16y+7 on both sides. Then we get the final answer.

0=0

Therefore all real numbers are possible solution to the given equation.

Math problem 2:

Solve p: 2p – 4q = 9

Solution:

Given 2p– 4q=9

If we can find p intercept value we set the q=0.

Then we can find the p value.

Substitute the value of q (q=0) into the above equation.

2p– 4(0) =9

2p– 0=9

2p=9

Divide by the value 2 on both sides. Then we get the final answer.

`(2p)/2` =`9/2`

p=`9/2`

Answer: `9/2`

Math problem 3:

Find the area of following figure:

Solution:

Given length= 15 ft and width= 7 ft

We can find the area of rectangle by using the following formula:

Area A= length*width or (l*w)

Substitute the value of l and w into the above formula and then we get the final answer.

Area A= 15*7

= 105 ft2

Answer: 105 ft2

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Practice problems of math:


Problems:

Solve x: 10+5x=50
Solve y: 2y+3x=5
Find the perimeter of square with the side value 5 cm.
Answer key:

x= 8
y=`5/2`
20 cm

Superstar Math Answers

Introduction to superstar answers:

The fundamental concepts of algebra, number, measurement, and geometry. Simple addition, subtraction, multiplication, and conservation are performed in the number. The Superstars program is to provide extra challenge that self- motivated to the students. More number of way to solve any problem. But student wants to know the shortcut methods. This shortcut method is one of the best way to study the math’s and easily understand the concept.


Math superstar answers:


Superstar math problem 1:

Rohit saw 4 cow and 2 dogs outside. How many legs did he see?

Solution:                          4 cow = 16 legs

2 dogs = 8 legs

8 + 16 = 24 legs

Answer:  24 legs

Superstar math problem 2:

Nijam have 5 nickel and 3 pennies, how much money do he have all together?

Solution:                   1 nickel = 5 pennies

So, 5 nickel = 25 pennies

Total = 25 + 3 = 28 pennies

Answer:  28 pennies

Superstar math problem 3:

Reena saw eight triangles, how many sides are there all together on eight triangles?

Solution:                                 1 triangle has 3 sides

8 triangle has 8 × 3 = 24 sides

Answer:  24 sides

Superstar math problem 4:

There are 5 cars and 6 tricycles in the neighbor's garage. How many wheels are there in all? Forget about any "spare tires"!

Solution:             20 wheels on the 5 cars,  and 18 on the 6 tricycles

Total = 20 + 18 = 38

Answer:  38wheels

Superstar math problem 5:

Martin has 16 balls, half of them are red. How many are Red?

Solution:                                  Total balls = 16

Half of ball = `16/2` = 8

Answer: 8 Red balls

Superstar math problem 6:

If the 22th day of the month falls on Thursday, on what day did the 11th fall?

Solution:   The calendar or make a chart with “Su, M, T, W, Th, F, Sa” at the top and begin numbering

backward putting 24 under Saturday. The date 17th and 10th must be fall on Saturdays and count Forward one

from the 10th.

Answer:  Sunday

Superstar math problem 7:

The sum of 3 consecutive numbers is 555. What are the numbers?

Solution:   We can use guess-check-revise method. Some students know that the numbers they seek are about `1/3`

of the total, and approximate the numbers by dividing 555 by 3. This gives 185, which is the middle number.

Answer:  184, 185, 186

Superstar math problem 8:

How many corners are on 3 triangles and 2 rectangles?

Solution:                                      1 triangle = 3 corners

1 rectangle = 4 corners

So, 3 triangle and 1 rectangle = (3 * 3) + (2 * 4) = 9 + 8 = 17 corners

Answer: 17 corners

Superstar math problem 9:

Together, 12 cars and 6 bikes weigh 1050 pounds. The bikes all weigh the same -- x pounds. Each cars weighs 55 pounds.

What is the  weigh of one bike?

Solution:      First finding the total weight of the 12cars: 12 × 55 = 660 pounds. Then they

will calculate 1050 - 660 = 390 pounds, the weight of the 6 bikes = 390 pounds.

So weight of one bike = ` 390 / 6` = 65 pounds per bike

Answer:  65pounds

Superstar math problem 10:

Find the side of the square if the perimeter of square is 20 m.

Solution:            The perimeter formula for square = 4a

4a = 20

a = ` 20/4` = 5 m

Answer: The side of the square 5m