Composition and Invertible Function

An invertible function for ƒ is a function from B to A, with the property that a round trip (a composition) from A to B to A returns each element of the first set to itself. A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1. Function composition is the applications of one function to the results of another. For instance, the functions f: X → Y and g: YZcomprised by computing the output of g when it has an argument of f(x) instead of x.  (Source. Wikipedia) .

Examples for invertible function:

To define the invertible of a function f i.e. f−1 (read as ‘f inverse’), the function f must be one-to-one and onto.

Example 1:

Let A = {1, 2, 3}, B = {a, b, c, d}. Consider a function f = {(1, a), (2, b),(3, c)}. Here the image set or the range is {a, b, c} which is not equal to the co domain {a, b, c, d}. Therefore, it is not onto.


For the inverse function f−1 the co-domain of f becomes domain of f −1.

i.e. If f : A → B then f−1 : B → A . According to the definition of domain, each element of the domain must have image in the co-domain. In f−1, the element‘d’ has no image in A. Therefore f −1 is not a function.

Example 2:

f = {(1, a), (2, a), (3, b)} where A = {1, 2, 3}, B = {a, b}

Here the two different elements ‘1’ and ‘2’ have the same image ‘a’.

Therefore the function is not one-to-one.

The range = {a, b} = B. ∴ The function is onto.

f(1) = a

f (2) = a

f(3) = b

Here all the elements in A has unique image

f −1 (a) = 1

f −1 (a) = 2

f −1 (b) = 3


The element ‘a’ has the two images 1 and 2. It violates the principle of the function that each element has a unique image. This is because the function is not one-to-one.

Thus, ‘f −1 exists if and only if f is one-to-one and onto’.

Examples for composition function:

Example :

Let A = {1, 2}, B = {3, 4} and C = {5, 6} and f : A → B and g : B → C such that f(1) = 3, f(2) = 4, g(3) = 5, g(4) = 6. Find gof.

Solution:

gof is a composition function from A → C.

Identify the images of elements of an under the composition function gof.

(gof) (1) = g(f(1)) = g(3) = 5

(gof) (2) = g(f(2)) = g(4) = 6

i.e. image of 1 is 5 and image of 2 is 6 under gof

∴ gof = {(1, 5), (2, 6)}

How to Calculate Median

To find the median in a set of data, intial step is rearrange the set of data in ascending order or descending order. The mid-term value is the median for thae set of data.Half the numbers in the record are fewer, and half the numbers are greater. Establish the Median, put the numbers are known in value arrange and locate the middle number. But there are two middle numbers (as happens when there is an even amount of numbers) then average those two numbers.


Single middle number:

Example 1:

Establish the median for the following listing of value:

8, 13, 4, 7, 5

Solution:

Find the Median of: 8, 13, 4, 7, and 5 (Odd amount of numbers)

Line up your numbers: 4, 5, 7, 8, and 13 (smallest to largest)

The Median is: 7 (The number in the middle)

Example 2:

Establish the median for the following listing of value:

3, 8, 3, 4, 3, 6, 4, 1, 3

Solution:

Find the Median of: 3, 8, 3, 4, 3, 6, 4, 1, and 3 (Odd amount of numbers)

Line up your numbers: 1, 3, 3, 3, 3, 4, 4, 6, and 8 (smallest to largest)

The Median is: 3 (The number in the middle)


Double middle numbers:

Example 1:

Establish the median for the following listing of value:

8, 3, 4, 7, 2, and 6

Solution:

Find the Median of: 8, 3, 4, 7, 2, and 6 (Even amount of numbers)

Line up your numbers: 2, 3, 4, 6, 7, and 8 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (4 + 6) / 2

= 10 / 2

= 5

The Median is 5.

Example 2:

Establish the median for the following listing of value:

8, 8, 8, 9, 9, 9, 11 and 10

Solution:

Find the Median of: 8, 8, 8, 9, 9, 9, 11 and 10 (Even amount of numbers)

Line up your numbers: 8, 8, 8, 9, 9, 9, 10 and 11 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (9 + 9) / 2

= 18 / 2

= 9

The Median is 9


Example 3:

Establish the median for the following listing of value:

7, 8, 6, 9, 8, 7, 12 and 11

Solution:

Find the Median of: 7, 8, 6, 9, 8, 7, 12 and 11 (Even amount of numbers)

Line up your numbers: 6, 7, 7, 8, 8, 9, 11 and 12 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (8 + 8) / 2

= 16 / 2

= 8

The Median is 8

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Learn Linear Functions Slope

A linear function is defined as the polynomial function contains the degree of one (y = mx + b). One can learn the linear equation relates a dependent variable with an independent variable in a simple way. The power of the linear function which is not always greater than one where there is no independent variable. A simple linear function with one independent variable (Ax + By + C = 0) traces a straight line when plotted on a graph. It is also called as linear equation. Learning the concept of slope using linear equations is known as learning linear functions slope.

Please express your views of this topic Linear Approximation Equation by commenting on blog.

Learning Linear Functions Forms:

The function is defined by,

f = { ( X, Y)/ Y = mX + b }

where m and b are constants, x and y is called a linear functions. The function derives a straight line while graphing.

Functions such as these gives graph that are straight lines, and, thus, the name linear. Linear functions come in three main forms.

Point Slope Form is given by the equation, m = (y - y1) / ( x – x1)
Slope-Intercept Form is given by the equation, y = mx +b
General Form is given by the equation, Ax + By + C = 0.


Slope of the Linear Functions Learning:

Calculations of rate at which the change takes place can be done under the concept of slope. Slope calculates the rate of change in the dependent variable as the independent variable changes. The slope is denoted by m.

Consider the linear function:

y = mx + b

where, m is the slope of the line and b is the y-intercept. Slope is defined as the ratio of unit change in y to the change in x.

slope m = Change in y / Change in x

=> m = (y_(2) - y_(1))/(x_(2) - x_(1))

Learning Linear Function Slope - Examples:

Find the slope of the line segment relating the following points:
(-1,-2) and (1, 6)

Sol:

Here, x1 = -1        

y1 = -2

x2 = 1        

y2 = 6

slope m = (y2 – y1) / (x2 – x1)

=> m = (6 – (-2) / (1 – (-1))

=> m = (6 + 2) / (1 + 1)

=> m = 8 / 2

=> Slope m = 4


Find the slope of the equation, 9x - 3y = 6

Sol:

9x - 3y = 6

=> -3y = 6 – 9x

=> y = (-1/3)(6 – 9x)

=> y = 3x – 2

It is in the general form, y = mx + b

Therefore, slope m = 3 and y-intercept b = -2.

Continuity Learning

Let 'a' be an aggregate and a in A, A function f : A |-> R is said to be continuous at "A". If each E > 0 EE   delta >0 such that x in A, || < delta  x - a

rArr   |  f ( x ) - f ( a ) | < E
lim_(x->a) f ( x ) = f ( a )
f is continuous at a    hArr lim_(x->a) f ( x ) = f ( a )

A function f : A |-> Ris said to be continuous on the aggregate 'a'. If ' f ' is continuous at every point a in A
A function f : A -> R is said to be discontinuous at a A. If ' f ' is not continuous at ' a '.


Continuity on the left and right at 'a':-

Let 'a' be an aggregate and a in A. A function f : A|-> R is said to be continuous on the left at 'a'. If each E>0 EE delta >0 such that x in A , a - delta < x <= a  rArr | f ( x ) - f ( a ) | < E

lim_(x->a-)  f ( x ) = f ( a )

Let 'a' be an aggregate and a in A. A function f : A|-> R is said to be continuous on the left at 'a'. If each E>0 EE delta >0 such that x in A , a <= x < a + delta   rArr | f ( x ) - f ( a ) | < E

lim_(x->a+) f ( x ) = f ( a )

Continuity of a function at open and closed intervals:-

A function ' f ' is said to be continuous on an open interval ( a, b ). If ' f ' is continuous at ' x ' AA x in ( a, b ).

A function ' f ' is said to be continuous on an closed interval [ a, b ].

' f ' is continuous at 'x' AA x in ( a, b )
' f ' is right continuous at 'a'.
' f ' is left continuous at ' b'.


Types of discontinuity:-

Let f : A -> Rand a in A be a point of discontinuity of ' f '.

' f ' is said to have Removable discontinuity at 'a'. If lim_(x->a) f ( x ) exists and lim_(x->a) f ( x ) != f ( a ) or f ( a )  is not defined.
' f ' is said to have Jump discontinuity or discontinuity of first kind at 'a'. If lim_(x->a-) f ( x ), lim_(x->a+) f ( x ) both exists and

lim_(x->a-)    f ( x )        !=  lim_(x->a+) f ( x )

' f ' is said to have sample discontinuity at 'a'. If ' f ' has removable discontinuity ( or ) jump discontinuity at 'a'.
' f ' is said to have finite discontinuity at 'a'. If ' f ' is not continuous at 'a' and ' f ' is bounded at 'a'.
' f ' is said to have Infinite discontinuity at 'a'.

If lim_(x->a) f ( x ) = oo   ( or ) lim_(x->a) f ( x ) = -oo
f ( x ) is unbounded in every neighbourhood of 'a'.

Example problems on Continuity

1)Examining the continuity of ' f ' defined by f ( x ) = | x | + | x - 1|

Solution:-    | x| = x if x > 0
= - x if x <= 0
| x - 1| = x if x > 1
= - x if x <= 1

If x <= 0
f ( x ) = | x | + | x - 1|
= - x - ( x -1 )
= 1 - 2x

If 0 < x < 1
f ( x ) =  | x | + | x - 1|
= x  - ( x - 1 )
= x - x + 1
= 1

If x >= 1
f ( x ) = | x | + | x - 1|
= x + x -1
= 2x -1

f ( x ) = 1 - 2x ; If x <= 0
= 1        ;  if 0 < x < 1
= 2x -1  ; if x >= 1

Continuity of ' f ' at x =0

Left Hand Limit            lim_(x->0-) f ( x ) = lim_(x->0-) 1 - 2x = 1

Right Hand Limit          lim_(x->0+) f ( x ) = lim_(x->0+) 1  = 1

f ( x ) = 1 - 2x if x <= 0
f ( 0 ) = 1

lim_(x->0) f ( x ) = f ( 0 ) = 1

Continuity of ' f ' at x =1

Left Hand Limit             lim_(x->1-) f ( x ) = lim_(x->1-) 1  = 1

Right Hand Limit            lim_(x->1+) f ( x ) = lim_(x->1+) 2x - 1  = 1

f ( x ) = 2x -1 if x >= 1
f ( 1 ) = 2 ( 1 )  - 1 = 1

lim_(x->1) f ( x ) = f ( 1 ) = 1

The function is continuous at x = 1