Math Homework Help

The function y= f(x) is defined by

is called a modulus function. It also known as absolute value function.

We can see that the domain of the modulus function is the set R of real numbers. and the range is the set of all non-negative real numbers.

That is

The graph of this function consists of

1. line y= -x, when x<0
2. point (0,0)
3. line y=x, when x >0

It is clear from the graph that is a many-one into function.

The function that associates each real number to itself is called the identity function and is usually denoted by I.

So, the function f:R->R defined by f(x)=x for all x in R is called the identity function.

From the knowledge of coordinate geometry, y=x represents a straight line passing through the origin and inclined at angle 45° with the x axis.

Clearly the domain and range of the identity function are both equal to R.

We can observe that it is a bijection.

A function of the type y=f(x)=k where k is a fixed real number.

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The graph of the constant function is a straight line parallel to x axis, which is above or below according to k is positive or negative. That is if k>0 the graph will be above x axis and at a distance k units above it. If k<0, then it will be k units below it. If k=0, then the graph will coincide with the x axis.

The domain of the constant function f(x)=k is the set R of all real numbers and range of the function is the singleton set {k}

So, we can see a constant function is a many-one into function.

The two cylindrical pans are similar. The diameter of the smaller pan is equal to the radius of the larger pan. How many of these smaller cans could fill the larger can?

Hint: Since the two cylinders are similar, their dimensions are in the same ratio. It is given that the diameter of the smaller pan is same as the radius of the larger pan. That is the radius of the two pans are in the ratio 2:1. In other words we can say if the radius of the larger pan is ‘r’ the radius of the smaller pan is r/2 and since they are similar their heights are also in the same ratio 2:1. So, if h is the height of the larger triangle, h/2 is the height of the smaller triangle.

Now,

Volume of the larger cylinder V_l= pi r²h

Volume of the smaller cylinder V_s=pi(r/2)²(h/2) =(pir²h )/8 = V_l/8

i.e V_s =V_l /8

Therefore 8 smaller pans can fill one larger pan.

The following figures represent a relationship between two variables.

Which rule relates x the number of dark squares to y, the number of white squares?

Ans:

y=2x+3

Sarah went to a grocery shop. There was a sales chart.

Sarah bought 4 carton potatoes and 1 cabbage. If she gave $10, how much money she got back?

Hint: 10-{(4×2.05)+.9}

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

It is very important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus.

Definition

Let f(x) be a function of x. Let a and l be constants such that as , we have . In such case we say that the limit of the function f(x) as x approaches a is l. we write this as

In case , no such number l exist, then we say that does not exist finitely.

Illustration

Let a regular polygon of n sides be inscribed in a circle. The area of the polygon cannot be greater than the area of the circle., however large the number of sides of the polygon increases indefinitely the area of the polygon continually approaches the area of the circle. Thus the difference between the area of the circle and the polygon can be made as small as we please by sufficiently increasing the number of sides of the polygon.

We have (Area of the polygon of n sides)=Area of the circle.

If an event can happen in exactly m ways, and if following it, a second event can happen in exactly n ways, then the two events in succession can happen in exactly mn ways.

Illustration.

Suppose there are five routs from A to B and three routs from B to C. In how many ways a person can go from A to C?

Since there are five different routs from A to b, the person can go the first part of his journey in 5 different ways. Having completed in any one of the 5 different ways , he has 3 different ways to complete the second part of the journey fro B to C. Thus each way of going from A to B give rise to 3 different ways of going from B to C.

There fore the total number of ways of completing the whole journey = number of ways for the first part x number of ways for the second part.
= 5 x 3=15.

Generalisation

If an event can occur in m different ways, a second event in n different ways, a third event in exactly p different ways and so on, then the total number of ways in which all events can occur in succession is mnp….

One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called central value or an average or the expected value of the variable.

The term ” average ” has been defined by various authors. Some important definitions are given below.

• “Average is an attempt to find one single figure to describe whole of figures” -Clark

• ” An average is a single value selected from a group of values to represent them in some way-a value which is supposed to stand for whole group, of which it is a part, as typical of all the values in he group” -A.E Waugh

• ” An average is a typical value in the sense that it is sometimes employed to represent all the individual values in a series or of a variable” -Ya-Lun-Chou

• “An average is a single value within the range of the data that is used to represent all the values in the series.Since an average is somewhere within the range of the data. It is also called a measure of central value” - Croxton &Cowden

I had been busy with some personal matters for the last five six weeks. In between gone for a vacation as well. But the new year day was a bit dull as my kids and I was not well after coming back from the vacation.

I wish you all a very prosperous and happy new year.