Study Domain and Range

The study about domain or input of a function f(x) is the total set of possible values of the independent variable in the function. The domain can also be given explicitly. The domain should be always x-values, and the range should be always y-values. The domain has not influenced by anything because it is an independent variable.

The study about range of a function is the total set of all feasible consequential values of the dependent variable of a function. The total set of all possible resulting values of the the range of function is the dependent variable of a function, after we have substituted the values in the domain.


Examples on study domain and range



Ex:1 find the domain and range of the following relation.
{(1, –4), (6, 5), (4, –1), (2, 6), (5, 3)}

Sol:

The domain should be always x-values, and the range should be always y-values. Therefore domain and range of given function is

Domain= {1, 2, 4, 5, 6}

Range= {–4, –1, 3, 5, 6}

Ex:2 State the domain and range of the following relation.
{(–4, 2), (5, 3), (–1, 1), (0, 5), (7, 5), (8, 5)}

Sol:

The domain should be always x-values, and the range should be always y-values

{(–4, 2), (5, 3), (–1, 1), (0, 5), (7, 5), (8, 5)}

domain:  {–4, –1, 5, 7, 8}

range:  {1,2,3,5}


STUDY DOMAIN AND RANGE OF TRIGONOMETRY FUNCTION


Ex:1 Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25}, Find the Domain and Range.

Consider the rule f: A → B: f (x) = x2 for all x Є A.

Then, each element in A has its unique image in B. So, f is a function from A to B.

f (1) = 12 = 1, f (2) = 22 = 4, f (3) = 32 = 9, f (4) = 42 = 16.

Domain(f) = {1,2,3,4} = A, co-domain(f) = {1,4,9,16,25} = B and range(f) = {1,4,9,16}.

Clearly, 25 Є B does not have its pre-image in A.

Ex:2 Let N be the set of all natural numbers. Find the domain and range for the function f(x)=2x.

Let f: N→ N: f(x) = 2x for all x fit in to N

Then, every element in N has its unique image in N.

So, f is a function from N to N.

Clearly f(1)= 2,f(2) = 4,f(3) = 6……., and so on.

Domain(f) = N, Co-domain(f) = N, Range(f) = {2, 4, 6, 8, 10}.

what is permutation math term

In math, the term permutation is the process of rearranging the given number of elements or objects. For example set {a,b,c), namely [a,b,c], [a,c,b], [b,a,c], [b,c,a], [c,a,b], and [c,b,a].

Formula for finding permutation: P(n,r) = `(n!) / ((n-r)!)` , where, n gives number of things, and r gives number of times.

Using permutation we can find how many possible ways are there to arrange the collection of objects. Permutation avoids the repletion. In this article we will discuss about the math term permutation and how to find the permutation.


Math Term Permutation – Example Problems


Example 1: How many different ways can a set of five country flags are arranged?

Solution:

P(5,5) = `(5!)/((5 - 5)!)` = `(5 * 4 * 3 * 2 * 1!) / (0!)` = 120              [0! = 1]

Therefore 120 possible ways are there to arrange a five flags.

Example 2: In how many ways 5 chocolates can be chosen from among 9 different kinds of chocolates?

Solution:

This problem involves 10 candies, taken 5 at a time.

P(10,5) = `(10!) / ((10 - 5)!)` = `(10 * 9 * 8 * 7 * 6 * 5!) / (5!)` = 30240

There are 30240 possible ways to choose 5 chocolates among 10 chocolates.

Example 3: Peter bought four movies. In how many different ways can he watch the four movies?

Solution:

P(4,4) = `(4!) / ((4 - 4)!)` = `(4 * 3 * 2 * 1) / (0!)` = 24                [0! = 1]

In 24 different ways he can watch the four movies.


Example 4: The computer password has 4 digits, if the possible digits are 2, 4, 5, 6, 7, 8, 9. How many different passwords can be made?

Solution:

This problem involves 7 digits, 4 digits at a time.

P(7,4) = `(7!) / ((7 - 4)!)` = `(7 * 6 * 5 * 4 * 3!) / (3!)` = 840

Therefore, 840 different passwords can be made.


Math Term Permutation – Practice Problems


Problem 1:  How many different ways can a set of seven country flags are arranged?

Problem 2: In how many ways can 6 candies chosen from among 9 different colors of candies?

Problem 3: Anita bought six movies. In how many different ways can she watch the six movies?

Answer: 1) 5040 2) 60480 3) 720

Learning High School Algebra

Algebra is a branch of mathematics that deals with  the study of  rules of operations and relations. Diophantus is regarded as the father of algebra. Basic algebraic concepts include variables and constants, expressions, terms, polynomials, equations and algebraic structures. So much is known about the subject that study of this takes a life time. Algebra offers lot of food for thought for students and mathematics lovers will have lot of fun solving problems in algebra. The beauty of the subject is that it prompts the student to think indicatively to solve problems and helps to develop analytical and logical skills for the student. Study of mathematics at any level will cover some topics in algebra.

High school algebra covers topics such as polynomials, algebraic expressions and identities, equtaions and factorization.

Let us learn some solved problems in the above topics of high school algebra.


Learning Polynomials and Factorization in high school algebra:


At high school level, polynomials and factorization are important topics covered in algebra.

POLYNOMIALS:

An Algebraic expression of the form axn is called Monomial in x. For example, 7x3   .The sum of two monomials are called a Binomial and the sum of three monomials are called Trinomial. For example, 2x3 + 3x is a binomial and 2x5 – 3x2 + 3 is Trinomial. The sum of a finite number of monomials in x is called a polynomial in x.

Example:

Find the sum of 2x4 – 3x2 + 5x + 3 and 4x + 6x3 – 6x2 – 1.

Solution:

Using the associative and distributive properties of real numbers, we obtain

(2x4 – 3x2 + 5x + 3) + (6x3 – 6x2 + 4x – 1) = 2x4 + 6x3 – 3x2 – 6x2 + 5x + 4x + 3 – 1

= 2x4 + 6x3 – (3+6)x2 + (5+4)x + 2

= 2x4 + 6x3 – 9x2 + 9x + 2.  (answer)

FACTORIZATION :

The process of writing polynomial as a product of two or more simpler polynomials is called Factorization.

The way of writing a polynomial as a product of two or more simpler polynomials is called factorization. The process of factorization is also known as the resolution into factors.

Example 1: Factorize x2 – 2xy – x + 2y.

Solution:

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

= x(x – 2y) + (–1) (x – 2y)

= (x – 2y) [x + (–1)]

= (x – 2y) (x – 1).   (answer)


Example 2: Solve 9x2 = x.

Solution:

9x2 = x

9x2 − x = 0

x (9x − 1) = 0        ab = 0

x = 0      or    x =

What if we attempt the same problem using the another method?

9x2 = x

9x = 1 divide by x

x =

Every quadratic equations has 2 roots. Dividing the quadratic equation by x removes the root x = 0.

However, dividing by a constant does not impact the roots.

Learning Algebraic Identites in high school algebra:

Algebraic Identities:

Algebraic identities is important in High school algebra . Algebraic identities are algebraic equations satisfied by any value of the variables.
The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original "anything," no matter what rate that "anything" (x) may be. Like normal algebra, Boolean algebra has its own unique identities base on the bivalent states of Boolean variables.

Example 4: Solve for variables x,y and z:

x + y + 2z = 2  ------> (1)

3x - y + 3z = 4  ------>(2)

2x + y + 4z = 6  ------>(3)

Solution:

Solve (1) and (2),

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

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

add the above two equations.

we get 4x + 5z = 6    ------> (4)

solve (1) and (3)

(3) * 2 ---->        4x + 4y +8z = 12

(4)     ----->        4x + 0y +5z = 6
(-)           (-)      (-)

3z = 6

z = 2

substitute z = 2 in (4) eqn

4x + 5(2) = 6

4x = -4

x = -1

substitute x = -1, z = 2 in (1) eqn.

(-1) + y + 2(2) = 2

y -3 = 2

y = 5.

x  = -1, y = 5, z = 2.

Write a Fraction for the Point

Fraction:

Fraction is defined as an element of quotient field. Fraction can be represented as `x/y` where fraction variable 'x' denotes the value called as numerator and fraction variable 'y' denotes the value called as denominator and the denominator 'y' is not equal to zero. It is used to write the fraction format for the given point.

Thus the fraction is classified as follows,

• Simple fraction
• Proper fraction
• Improper fraction
• Complex fraction

write a fraction for the point : Types of fractions


Simple fraction:

Simple fraction is a fraction, which has both numerator and denominator as whole number.

Ex:

`1/5` , `2/7` , `8/9`

Proper fraction:

It is a fraction, which has a numerator less than its denominator, and the value of that fraction is less than one.

Ex:

`3/5` , `1/8` , `24/25`

Improper fraction:

Improper fraction is a fraction, where the top number of fraction that the numerator is greater than or equal to its own denominator (bottom number) and the value of that fraction is greater than or equal to one.

Ex:

`7/2` , `45/23` , `123/120`

Complex Fractions:

If a fraction of numerator and denominator contains a fraction, it is called complex fraction.

The complex fraction is also called as a rational expression because it has a numerator and denominator with fraction. Otherwise, the overall fraction includes at least one fraction.

Ex:

` (7/3) / (4/5)`


Example problems for write a fraction for the point:


Ex 1

Write the fraction for the following point: (0.5, 1.5, and 0.88)

Sol:

0.5

Step 1 :( multiply  and divide by 10 on both sides, we get )

`(0.5)*(10)/10`

= `5/10`

Step 2: simplifying we get

=`1/2`

like wise for the following numbers we get,

1.5 = `3/ 2 ` ( multiply and divide by 2 on both sides)

0.88 = `8/9 ` ( multiply and divide by 10 on both sides)

Ex: 2

Write the Equivalent fraction for the following points: (0.25, 0.75, 2.5, and 50)

Sol:

0.25 = `1/4 ` = `2/8` = `3/12 `

0.75 = ` 3/ 4` = `6/8` = `9/12`

2.5 = `5/2` = `10/4` = `15/6 `

50.0 = `100/2` = `200/4 `

study graph numbers

Number:

     A number is defined as a numerical thing, which is used for measuring and counting. It is also called as numeral and includes zero, negative numbers, rational numbers, irrational numbers, etc,. The procedure of numerical operation involves one or more numerical as input and generate its relevant numerical output. This operation includes arithmetic function such as addition, subtraction, multiplication, division, and exponentiation.

Study about graph numbers:

Composite Number:

A Composite Number is a number, which can be divided with evenly. The composite number has additional than two factors with one and itself.

Otherwise, Composite number can be defined as numeral (integer) that is accurately divided with minimum one factor except one and itself. Composite number has infinite numbers also. However, composite numbers are not prime numbers.

Prime Number:

A prime number is a number, which can be divided only with one and itself. The prime number has only two factors such that one and itself.

Cardinal numbers:

Cardinal number is defined as counting numbers in words, which is representing the quantity such as five dogs, three boys. Otherwise, it is referred to as overview of natural numbers, which is calculating the cardinal (size) of the given sets.

Ordinal number:

     Ordinal numbers are numbers, which denote the order or location of objects with number words in a series such as ‘first’, ‘second’, ‘third’... It is also called as ordinals.

Graphing numbers:

     Graphing numbers is the graphical representation of integers with inequalities in horizontal line. It is a visualizing result of number line with simple steps.

Inequality:

     Inequality is defined as two real numbers or two algebraic expressions are related with functioning a sign as ‘<’ (less than), ‘>’ (greater than), ‘≤’ (less than or equal) and ≥ (greater than or equal).

Examples for graph:

Example 1) graph the following points

A= 3 + 2i
B =  -4 + 5i
 C = -5 - 4i
 D = i

Solution: 

2) Graph the following inequalities:

a) -1≤w≤4

Solution:

The value of ‘w’ is -1, 0, 1, 2, 3, 4

So graphing of ‘w’ on number line is

 

b) 1≤q<5

Solution:

The value of ‘q’ is 1, 2, 3, 4

So graphing of ‘q’ on number line is

Study Exponentiation

In math, exponentiation is the operation, which is written as the form of an. Where a and n is said to be base and exponent and n is any positive integer. Normally, exponentiation means that repeated multiplication. Otherwise, exponentiation an is the product of n factors of a. The exponent is usually placed as a superscript to right of base value. We are having many properties for exponentiation. Let see properties and example problems for exponentiation.


Properties - Study Exponentiation


We are having seven number of exponentiation properties that used for solving problems with exponentiation. In this properties, a, m and n are any integer values.

Product of like bases:

am an = am+n

Quotient of like bases:

`a^(m)/a^(n)` = am-n

Power to a power:

` (a^(m))^(n)` = amn

Product to a power:

(ab)m = am bm

Quotient to a power:

`(a/b)^(n)` = `a^(n)/b^(n)`

Zero exponent:

a0 = 1

Negative exponent:

a-n = `1/a^(n)` or `1/a^(-n)` = an

These are the properties that are used for exponentiation problems in study math.


Example Problems - Study Exponentiation


Example 1:

Solve 23 22.

Solution:

Given, 23 22.

This is in the form of am an, so we need to use am an = am+n property.

Here, m = 3 and n = 2 and a = 2.

Thus, 23 22 = 23+2

= 25

= 2 × 2 × 2 × 2 × 2

= 32
Hence, the answer is 23 22 = 32.

Example 2:

Shorten the following `5^5/5^3`.

Solution:

Given, `5^5/5^3` .

This is in the form of `a^m/a^n` , so we need to use `a^m/a^n` = am-n property.

Here, m = 5 and n = 3 and a = 5.

Thus, `5^5/5^3` = 55-3

= 52

= 5 × 5

= 25
Hence, the answer is `5^5/5^3` = 25.

That’s all about the study exponentiation.

Solving Change of Base Formula

The solving change of base formula is known as formulas which it permits us to rework a logarithm by means of the logs that may be is written with different base.

The change of base formula is given by,

Log a x = log b x / log b a

Here, assume that a, b and x are positive where a≠1 and b≠1.

Importance in solving change of base formula:

• Using the change of base formula we can change any base to another base. The most commonly used bases are base 10 and base e.

Log a x = log b x / log b a

• The solving change of base formula is used highly if calculators to assess a log to several base further than 10 or e.
• At the solving change of base formula having the value of x which is superior than zero.
• The log of a number to a given base is the power or an exponent to which the base must be raised in order to produce that number.

Advantages  in using change of base formula:

• Change the numeral bases, like convert  from base 2 to base 10m which is known as base conversion.
• The logarithmic change-of-base formula is applicable regularly in algebra and calculus.
• It is used for varying among the polynomial and normal bases.

solving Examples using change of base formula:


1) Solve  log 816

Solution:

By solving the change of base formula

=> log a x = log b x / log b a

log 8 16 = log 2 16 / log 2 8

=  4 / 3

2) Solve log 918

Solution:

By solving the change of base formula

=> log a x= log b x / log b a

log 9 18 = log 2 18 / log 2 9

= 4.169 / 3.169

= 1.315

3) Solve log 2 5.

Solution:

By solving the change of base formula

=> log a x= log b x / log b

log 2 5 = log 10 5 / log 102

The approximate value of the above expression is solving by,

=0 .6989 / 0.30103

=0 .3494

Practice problems in solving change of base formula :

1) Solve log 3 9 using the change of base formula.

Answer: 2

2) Solve log 10 8 using the change of base formula

Answer: 0 .9030