Multiple Value Function

During mathematics, a multivalued function is an overall relation; specifically all input is related with one or else further outputs. Exactingly words, a well-defined function connections one, also only one, output toward at all particular input. The word "multiple value function" is, consequently, a misnomer while functions be single-valued.Multiple value functions frequently occur as of functions are not injective. Such functions perform not contain an inverse function, except they perform include an inverse relation. It is also called the set-valued function.

Examples for multiple value function

 

This figure do not symbolize a right function since the element 3 within x be related through two elements b with c within y.

Each real number larger than zero or else each complex numbers except for 0 have two square roots. The square roots of 9 be in the set {+3,−3}.

The square roots of 0 be describe through the multiset {0,0}, since 0 be a root of multiplicity 2 of the polynomial x².

Every complex number include three cube roots

Complex logarithm functions are multiple-valued. The values implicit with log (1) be 2πni for every integers n.

Inverse trigonometric function is multiple-valued since trigonometric function is periodic.

tan(π/4)=tan((5π/4)=tan((-3π/4)=tan(2(n+1)π/4)=...1

Therefore arctan(1) might be consideration of since contain multiple values such as π/4, 5π/4, −3π/4, and rapidly. We know how to treat arctan since a single-valued function through restrict the domain of to -π/2 < x < π/2. Therefore, the range of arctan(x) becomes -π/2 < x < π/2. These values as of a limited field are call principal values.

Example problems

Problem 1
Given function f(x)=7x-5, what is the value of f(1) and(f(2)?
solution:

Given function f(x)=7x-5)

We can  find the value of f(1), to substitute 1 for the given function,

f(1)=7x1-5

      =7-5

f(1)=2

We can  find the value of f(2), to substitute 2 for the given function,

f(2)=7x2-5

      =14-5

f(2)=9

Problem 2
Given function f(x)=5x²+6x-5, what is the value of f(5)?
solution:

Given function f(x)=5x²+6x-5

We can  find the value of f(5), to substitute 5 for the given function,

f(5)=5x5²+6x5-5

      =5x25+6x5-5

     =125+30-5

  f(5)=150

Rectangular Objects

Rectangular objects are one of the basis of mathematics. Rectangular objects are having six sides. One of the rectangular objects is cuboid. Cuboid are also having six sides. These rectangular cuboid are having same faces like the rectangular objects. Rectangular prism is also included in the rectangular objects. Length of the rectangular prism is equal to the length of the rectangular.

Explanation for rectangular objects

There are many rectangular objects are present. They are defined as,

1. Rectangular cuboid

2.Rectangular prism

Rectangular cuboid:

The volume of the rectangular cuboid are find using the formula that are shown below,

Volume = Height `xx` Width `xx` Length

This can also be written as,

V = h `xx` w `xx` l.

Surface area of the rectangular cuboid are find using the formula,

Surface area = 2wl `xx` 2lh `xx` 2hw

The diagrammatic representation of the cuboid are shown below,



Example problem for rectangular objects

Example problem 1: Find the surface area and volume of the cuboid by using the formula, where the height = 4, length = 3, width = 2.

Solution:

Step 1: The volume of the rectangular cuboid are,

Volume = Height `xx` Width `xx` Length

from given h= 4, l =3, w =2

Step 2: Therefore by substituting the given information, we get,

V = 4 `xx` 3 `xx` 2

= 24.

Therefore, the volume of the cuboid is 24units.

Step 3: The surface area of the cuboid is given by,

Surface area = 2wl `xx` 2lh `xx` 2hw

Step 4: by substituting the above information given, we get,

Surface area = 2 ( 3 `xx` 2 ) `xx` 2( 4 `xx` 3 ) `xx` 2 ( 3 `xx` 2 )

= 2(6) `xx` 2(12) `xx` 2(6)

= 12 `xx` 24 `xx` 12

= 3456

This is the required area for cuboid.

Example problem 2: Find the surface area and volume of the cuboid by using the formula, where the height = 3, length = 4, width = 5.

Solution:

Step 1: The volume of the rectangular cuboid are,

Volume = Height `xx` Width `xx` Length

from given h= 3, l =4, w =5

Step 2: Therefore by substituting the given information, we get,

V = 3 `xx` 4 `xx` 5

= 60

Therefore, the volume of the cuboid is 60 units.

Step 3: The surface area of the cuboid is given by,

Surface area = 2wl `xx` 2lh `xx` 2hw

Step 4: by substituting the above information given, we get,

Surface area = 2 ( 6 `xx` 4 ) `xx` 2( 4 `xx` 3 ) `xx` 2 ( 3 `xx` 6 )

= 2( 24 ) `xx` 2( 12 ) `xx` 2( 18 )

= 288 `xx` 24 `xx` 36

= 248832

This is the required area for cuboid.

Adding Fractions Grade Six

Fractions:

A fraction is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development was the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.(Source: Wikipedia)

This article will help for grade six students. Through this article grade six students learn about adding fractions.We are going to see some solved problems on adding fractions and some practice problems on adding fractions.

Solved problems on adding fractions grade six:

Problem 1:

Add the fractions 1/8 and 1/8

Solution:

Given, 1/8 + 1/8

Here both fractions have the same common denominator.

So we Can add the numerator normally and keep the denominator as it is.

1/8 + 1/8 = (1+1)/8

= 2/8

We can also simplify it further,

(2÷2)/(8÷2) = 1/4

Answer: 1/8 + 1/8 = 1/4

Problem 2:

Adding the fractions 1/8 + 1/16

Solution:

Given, 1/8  + 1/16

Both fractions have different denominator,

So we need to find least common denominator

Multiple of 8 = 8, 16, 24 ,32...

Multiple of 16 = 16 , 32 ....

Least common multiple = 16.

So to make a common denominator, multiply 1/8 by 2 on both  numerator and denominator,

(1 * 2) / (8 * 2) = 2/16

Now we can subtract,

1/8 + 1/16 = 2/16 + 1/16

= (2 + 1) / 16

= 3/ 16

Answer: 1/8 + 1/16 =3/16

Problem 3:

Adding the fraction 12 / 32 + 6 / 18

Solution:

Given , 12 / 32 + 6 / 18

Both fractions have different denominator,

So we need to find least common denominator

Multiple of  32 = 32, 64, 96, 128, 160, 192, 224 32, 64, 96, 128, 160, 192, 224, 256, 288, 320.........

Multiple of  18 = 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324........

The least common multiple of 32 and 18 is 288.

So we need to make the denominator as 288.

(12 * 9) / ( 32 * 9 ) = 108 / 288

( 6 * 16) / ( 18 * 16) = 96 / 288

12 / 32  + 6 / 18 = 108 / 288 + 96 / 288

= ( 108 + 96) / 288

= 204 / 288

= 51 / 72

Practice problems on adding fractions grade six :

Problems:

1. Adding the fractions  6/7 + 15 /7

2. Adding the fractions 3/4 + 5/6

Answer:

1.3

2.19 / 12

Probability and Chance

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, which 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. In this article we shall discuss about probability and chance problems.(Source: wikipedia)

Probability and chance example problem

Example 1:

If the odds in favor of an event be 3/5 find the chance of the occurrence of the event.

Solution:

Let the given event be E and let P(E)=x then

Odds in favor of E =`(P(E))/(1-P(E))`

= `(P(E))/(1-P(E))` =`3/5`

= `(x)/(1-x)` =`3/5`

5x=3-3x

8x=3

x= `3/8`

Therefore required probability = `3/8`

Example 2:

There dice are thrown together. Find the chance of getting a total of at least 6.

Solution:

In throwing 3 dice together the number of all possible outcomes is (6x6x6)=216

Let E = event of getting a total of at least 6.

Then, E = event of getting a total of less than 6.

= event of getting a total of 3 or or 5.

={(1, 1, 1),(1,1,2),(1,2,1),(2,1,1)(1,1,3),(1,3,1),(3,1,1),(1,2,2),(2,1,2)(2,2,1)}

Now , n(`barE` )=10= P(not E)=P(`barE` )=`(n(barE))/(n(S))` =`10/216` =`5/108`

=P(E)=1-P(not E)=1-`5/108` =`103/108`

Hence the required probability is `103/108`

Example 3:

A bag contains 12 red and 15 white balloon. One ball is drawn at random. Find the chance that the ball drawn is red

Solution

Total number of balls=(12+15)=27

Let S be the sample space. Then

n(S)= number of ways of selecting 1 balloon out of 27=27

Let E be the event of drawing a red balloon. Then

n(E)= number of ways of selecting 1 red balloon out of 12=12

Therefore P(getting a /red balloon)=P(E)=`(n(E))/(n(S))` =`12/27` = `4/9`

Probability and chance practice problem

Problem 1:

The odds in favor of the outcomes of an event are 8:13 find the chance that the event will occur

Answer:

`8/21`

Problem 2:

If the odds against the occurrences of an event be 4:7 find the chance of the occurrences of the event?

Answer:

`7/11`

Graph Systems of Inequalities

Algebra is a subdivision in mathematics in which comprises of infinite number of operations on equations, polynomials, inequalities, radicals, rational numbers, logarithms, etc. Graphing system algebra inequalities is also a part of algebra. It is similar to the graphing of ordinary equations, but here we got to graph the inequality given either greater or less than. In graphing system of inequalities , the area overlapping is the solution set.. The following sections helps us to learn much better about graphing system of inequalities.

Steps to solve system of inequalities:

The steps involved in graphing system of inequalities are as follows,

Consider the inequality y > ax+c
Step 1: Convert the given equation as y =ax+c.
Step 2: Since the given inequality is a function of x, let y =f(x).
Step 3: Therefore f(x) = ax+c.
Step 4: Substitute various values for ‘x’ and find corresponding f(x).
Step 5: Table the values as follows x  &  f(x)  the values of x as -3, -2, -1, 0,1,2,3. and for f(x) their corresponding values.
Step 6: The values in the table are the co-ordinates, graph them.
Step 7: Shade the inequality range above the line, since greater than symbol (>) is given.
Step 8: Shade the inequality range below the line, if less than symbol (>) is given.
Step9: Repeat the same steps for next equation also.
Step 10: Shade the region which is overlapped, which is the solution region for the system of inequalities.


Example problem for graphing inequalities:

Problem 1:

Graph the given system of inequalities and find the solution region,

3x - y < 4

2x + y < 3
Convert the given equation as

y >3x+4
Since the given inequality is a function of x,
Let y =f(x).
Therefore
f(x) = 3x+4.

Substitute various values for ‘x’ and find corresponding f(x).
When x= -3
f(-3) = (-3)3 +4,
-9+4,
-5, therefore the co-ordinates are (-3, -5)

When x= -2
f(-2) = (-2)3 +4,
-6+4,
-2, therefore the co-ordinates are (-2, -2)

When x= -1,
f(-1) = (-1)3 +4,
-3+4,
1, therefore the co-ordinates are (-1, 1)

When x= 0
f(0) = (0)3 +4,
0+4,
4, therefore the co-ordinates are (0, 4)

When x= 1
f(1) = (1)3 +4,
3+4,
7, therefore the co-ordinates are (1, 7)

When x= 2
f(2) = (2)3 +4,
6+4,
10, therefore the co-ordinates are (2, 10)
Tabulate and graph them, and mark the inequality
Following the same procedure for the second equation and graph them. the graph looks like,

On Graphing them combined, we get the solution region as ,

Problem 2:

Graph the system of inequalities

4x +3y <2

2x+y >8

Solution:

Following the above procedure and graphing them, the solution range is as follows,

Radical Rules Math

Radical symbol used to indicate the square root or nth root. Radical of an algebraic group, a concept in algebraic group theory. Radical of a ring, in ring theory, a branch of mathematics, a radical of a ring is an ideal of "bad" elements of the ring. Radical of a module, in the theory of modules, the radical of a module is a component in the theory of structure and classification. Radical of an ideal, an important concept in abstract algebra. The radical symbol is ' √ ' . The cubic root of a can be expressed as `root(3)(a)`   and nth root of x can be expressed as  ` rootn x `                                                                                                                                                                                                   

                                                                                                                                                                                                                   Source Wikipedia.

I like to share this sine rules with you all through my article. 

Radical rules in math:

                    Math radical has product rule, Division rule and exponential rule.
Math Product rules for radical:
              Square root product rule:         `sqrt(ab)`   =  ` sqrta * sqrt b`
              Cube root product rule:          `root3 (ab) `   = ` root3 (a) * root3 (b)`
              n^th root product rule                ` rootn (ab) = rootn (a) * rootn (b)`
Examples for radical product rule math problem 1:
              Multiply the two math radical `sqrt5 * sqrt14`
        Solution:
                            Given radicals`sqrt5 * sqrt14`        
               We know the math radical product rule  `sqrt(ab)`   =  ` sqrta * sqrt b` 
                                     So, `sqrt5 * sqrt14`  =`sqrt(5 * 14)`
                                                                = `sqrt70`
        Answer: `sqrt70` 
Examples for radical product rule math problem 2:
              Multiply the two math radical `sqrt 18 * sqrt 3`
        Solution:
                            Given radicals `sqrt18 * sqrt3`           
               We know the math radical product rule  `sqrt(ab)`   =  ` sqrta * sqrt b` 
                                     So,  `sqrt18 * sqrt3` =`sqrt(18 * 3)`
                                                                = `sqrt54`
        Answer:  `sqrt54`
Math Division rules for radical:
                Square root division rule:        ` sqrt(a/b)` = `sqrta / sqrt b`
               Cube root division rule:          ` root3 (a/b)` =  `root3 (a) / root3 (b)`
               n^th root division rule                 `rootn (a/b)` = `rootn (a) / rootn (b)`
Examples for radical division rule math problem 1:
              Simplify  the math radical ` root3 (66) / root3 (11)`
       Solution:
                            Given math radicals  ` root3 (66) / root3 (11)`      
               We know the math radical division rule    ` root3 (a/b)` =  `root3 (a) / root3 (b)`
                                                So, ` root3 (66) / root3 (11)`   = `root3 (66/11)`
                                                                      = `root3 6`
        Answer: `root3 6`
Examples for radical division rule math problem 2:
              Simplify  the math radical ` root4 (16) / root4 (24)`
       Solution:
                            Given radicals ` root4 (16) / root4 (24)`      
               We know the math radical division rule    `rootn (a/b)` = `rootn (a) / rootn (b)`
                                              So, ` root4 (16) / root4 (24)`   = `root4 (16/24)`
                                                                      = `root4 (2/3)`
        Answer:  `root4 (2/3)`

Other Radical rules in math:

Relation rules with exponential term:
                      `root(n)(x)`^m =( `root(n)(x)` )^m = (x^1/n )^m = x^m/n
Examples for radical with exponent math problem 1:
              Simplify the math radical `"(root5 4)^5 * `
        Solution:
                     Given math radicals `(root5 4)^5 * (sqrt27)^2`
                                                = `(root5 4)^5 * (sqrt27)^2`
              we know the math radical rule with an exponents  ( `root(n)(x)` )^m = x^m/n
                          So,     = 4^5/5 * 27^2/2
                                  = 4¹ * 27¹
                                              = 4 * 27
                                              = 108
        Answer:   108
Examples for radical with exponent math problem 2:
              Simplify the math radical `(root4 5)^3 * (sqrt5)^3`
        Solution:
                     Given radicals `(root4 5)^3 * (sqrt5)^3`
                                              = `(root4 5)^3 * (sqrt5)^3`
              we know the math radical rule with an exponents  ( `root(n)(x)` )^m = x^m/n
                          So,     = 5^3/4 * 5^3/2
                                  = `5` ^{(`3/4` + `3/2` )}
                                              = 5^9/4
             Answer:  5^9/4

Calculus Help with Assignment

Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

(Source: Wikipedia)

Example problems for calculus help with assignment

Calculus help with assignment problem 1:

Differentiate the given function with respect y and find the value of g'(y). The given function is g(y) = 0.45y3 + 3.3t2 - 4y4 + y5

Solution:

The given function is g(y) = 0.45y3 + 3.3t2 - 4y4 + y5

Differentiate the given function with respect to x, we get

g'(y) = (0.45 * 3)y2 + (3.3 * 2)y - (4 * 4)y3 + 4y4

= 4y4 - 16y3 + 1.35y2 - 6.6y

Answer:

The final answer is 4y4 - 16y3 + 1.35y2 - 6.6y

Calculus help with assignment problem 2:

Differentiate the given equation with respect to t, y = 23t5 + 42t3 - 9t2 - 16

Sol:

Given y = 23t5 + 42t3 - 9t2 - 16

Differentiating both sides, we have

`(dy / dt)` = (23 * 5)t4 + (42 * 3)t2 - (9 * 2)t - 0

= 115t4 + 126t2 - 18t

`(dy / dt)` = 115t4 + 126t2 - 18t

Answer:

The final answer is 115t4 + 126t2 - 18t

Calculus help with assignment problem 3:

Integrate the given function ∫ (9x2) dx

Solution:

The given function is 9x2 dx

Integrate the given function with respect to x, we get

∫ (9x2) dx = 9 `(x^3 / 3)` + c

= 3x3 + c.

Answer:

The final answer is 3x3 + c

Calculus help with assignment problem 4:

Integrate the given function g(x) and find the value of G(3). The given function is g(x) = 14x6 - 5x4 + 28x + 40

Solution:

The given function is g(x) = 14x6 - 5x4 + 28x + 40

Integrate the given function with respect to x, we get

G(x) = 14 `(x^7 / 7)` - 5 `(x^5 / 5)` + 28 `(x^2 / 2) ` + 40x + c

= 2x7 - x5 + 14x2 + 40x + c

Find the value of G(3):

Substitute x = 3 in the above equation, we get

G(3) = 2 (3)7 - (3)5 + 14 (3)2 + (40 * 3)

= 4374 - 243 + 126 + 120

= 4377

Answer:

The final answer is 4377

Practice problems for calculus help with assignment

Calculus help with assignment problem 1:

Find the value f'(2). Given function f(x) = 12x3 - 4x2 + 9

Answer:

The final answer is 128

Calculus help with assignment problem 2:

Integrate the given function g(x) = 6x2 + 12. Find the value G(4).

Answer:

The final answer is 176