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But you will see that 2 is less than any other upper bound. Hence 2 is the supremum or the least upper bound of T. You will agree that -1 is the 1. Note that for both the sets T and Z-, the 1. This may not be true in general. The 1. But 0 c R-. Working on similar lines we can also define a lower bound for a given set S to be a real number v such that v I x for all x E S.
We shall say that a set is bounded below, if we can find a lower bound for it. Further, from among all the lower bounds of a set S, which is bounded below, we can choose a lower bound v such that v is greater than or equal to every lower bound of S. It is easily seen that, if such a v exists, then it is unique.
We call this v the greatest lower bound or the infimum of S. As in the case of l. We shall say that a set S c R is bounded if it has both an upper bound and a lower bound.
Based on this discussion you will be able to solve the following exercise. Every nm-empty subset S of R that is bounded above, has a supremum. We shall use this property in Unit Many more properties are either restatements or consequences of these sixteen properties. Here is a list of some of them. Additive inverse is unique, i. Multiplicative inverse is unique, i.
Definitian 2: Now we are ready to list a few more properties. You are already aware of these. But let us quickly recall them. The set N of natural numbers. Note that it is the smallest subset o f R possessing the following properties: It is the smallest subset of R possessing the following properties:. The set Q of rational nhmbers. We observe that it is the smallest subset of R possessing the following properties:.
You must have also studied the following properties of these sets. They do not, however, satisfy A3, A4 and M4. Therefore Q is an ordered field. But C is not satisfied, that is, Q is not order-complete. We list here some more properties of these sets which you will find useful in our study of calculus: A real number s is the supremum of a set S c R if and only if the following conditions are satisfied. Now But Every nonempty set of real numbers that is bounded below, has an infmum.
What is its infimum? Give an example. In this section we shall define the absolute value of a real number. You will realise the importance of this simple concept as you study the later units. Definition 3: If x is a real number, its absolute value, denoted by I x I read as modulus of x, or mod x , is defined by the following rules:. The following theorem gives some of the ,. Let us consider these one by one.
Since I holds for all x and y in R, therefore, by interchanging x and y in 1 we have. Therefore,max 1x y 1,- 1x1-I y 1 I x - y l. But the left hand side of the above inequality is simply x - 1 y. Now you should be able to prove some easy consequences of this theorem.
The following exercise will also give you some practice in manipulating absolute values. This practice will come in handy when you study Unit 2. E 3 Prove the following: This means that the difference between x and a is not more than 6. Before we define an interval let us see what is meant by a number line.
The real numbers in the set R can be put into one-to-one correspondence with the points on a straight line L. In other words, we shall associate a unique point on L to each real number and vice versa.
Consider a straight line L [see Fig. Mark a point 0 on it. The point 0 divides the straight line into two parts. We shall use the part to the lefr of 0 for representing negative real numbers and the part to the right of 0 for representing positive real numbers. We choose a point A on L which is to the right of 0. We shall represent the number 0 by 0 and 1 by A. OA can now serve as a unit.
We thus find that to each real number we can associate a point on the units -: Further, z is positive if S is to the right of 0, and z is negative if S is to the left of 0. This representation of real numbers by points on a straight line is often very useful. Because of this one-to-one correspondence between real numbers and thg points of a straight line, we often call a real number "a point of R. Similarly L is called a "number line".
Note that the absolute value or the modulus of any number x is nothing but its distance from the point 0 on the number line. In the same way, 1 x - y 1 denotes the distance between the two numbers x and y [seeFig.
Now let us consider the set of the real numbers which lie between two given real numbers a and b, where a I b. Actually, there will be four different sets satisfying this loose condition. These are: Each of these sets is called an interval, and a and b are called the end points of the interval.
The interval ]a, b[, in which the end points are not included, is called an open interval. Note that in this case we have drawn a hollow circle around a and b to indicate that they are not included in the graph.
The set [a, b], contains both its end points and is called a closed interval. In the representation of this closed interval, we have put thick black dots at a and b to indicate that they are included in the set.
The sets [a, b[ and ]a, b] are called half-open or half-closed intervals or semi-open or semiclosed intervals, as they contain only one end point.
This fact is also indicated in their geometrical representation. Each of these intervals is bounded above by b and bounded below by a.
Yes, we can. We know that 1 x - a 1 can be thought of as the distance between x and a. This means I is the set of all numbers x, whose distance from a is less than 6. This means if x E 12, then the distance between x and a is less than 6, but is not zero.
We can also say that the distance between x and a is less than 6, but x a. As you can see easily, none of these sets are bounded. For instance, ]a, -[ is bounded below, but is not bounded above, 1- -, b] is bounded above, but is not bounded below.
We note further that if S is any interval bounded or unbounded and if c and d are two elements of S, then all numbers lying between c and d are also elements of S.
Now let us move over ro present some basic facts about functions which will help you refresh your knowledge. We shall look at various examples of functions and shall also define inverse functions. Let us start with the definition of a function.
Definition 4: If X and Y are two sets, a function f from X to Y, is a rule or a correspondence which connects every member of X to a unique member of Y.
We write f: We shall denote by f x that unique element of Y which is associated to x E X. Example 1: The domain here is N and the co-domain is R. Example 2: Example 3: Every natural number can be written as a product of some prime numbers. This rule does not associate a unique number with 6 and hence does not give a function from N to N. Thus, you see, to describe a function completely we have to specify the following three things: The rule which defines a function need not always be in the form of a formula.
But it should clearly specify perhaps by actual listing the correspondence between X and Y. The set of f-images of all members of X, i. It is easy to see that f X c Y. Remark 3 a Throughout this course we shall consider functions for each of which whose domain and co-domain are both subsets of R.
Such functions are often called real functions or real-valued functions of a real variable. We shall, however, simply use the word 'function' to mean a real function. The variable x or t or u is also called an independent variable, and f x is dependent on this independent variable. Graph of a function: A convenient and useful method for studying a function is to study it through its graph. To draw the graph of a function f: For each x E X, the orderedpair x, f x determines a point in the plane see Fig.
The set of all the points obtained by considering all possible values of x remember that the domain o f f is X is the graph of the function f. The role that the graph of a function plays in the study of the function will become clear as we proceed further. In the meantime let us consider same more examples of functions and their graphs.
The simplest example of a hnction is a constant function. A const'. In general, the graph of a constant function f: The graph of this function is shown in Fig. The domain in each case is R. Can you identify them? If a is a positive real number other than I , we call define a function f as:. This function is known as the exponential function. A special case of this flmction, where a c , is often found useful.
Its range is the set R'of positive real numbers.
The range of this function is R. Its grnp! Take a real number x. We say that n is the greatest integer not exceeding x, and denotc it by [x]. For example. I 3 and Let 11s consider the function defined on R by Yetrmg f x -- [XI. This function is called the greatest integer function. The graph of the function is as shown in Fig. It resembles the steps on an infinite staircase. Notice that the graph consists of infinitely many line segments of unit length, all parallel to the x-axis.
This is defined for all real x, for which k x O. In this sub-section we shall see what is meant by the inverse of a function. But before talking about the inverse, let us look at some special categories of functions. These special types of functions will then lead us to the definition of the mverse of a function. One-one and Onto Functions Consider the function h: Thus 2 and -2 are distinct members of the domain R, but their h-images are the same.
Can you find some more numbers whose h-images are equal? This may be expressed by saying that 3x, y such that? Now, consider txe fuilction g: For,x, x,. While one of them, namely g, sends distinct members of the domain to distinct members of the co-domain, the other, namely h, does not always do so. We give a special name to functions like g above. Definition 5: A function E x x Y is said to be a one-one function a 1 - 1 function or an injective function if the images of distinct members of X are distinct members of Y.
Thus the function g above is one-one. Remark 4: The function h: On the other hand, the function g: This shows that every member of the co-domain is a g-image of some member of the domain and thus.
The following definition characterises this property of the function. Definition 6 A function f: X x Y is said to be an onto function or a surjective function if every member oEY is the image of some member of X. Thus, h is not an onto function, whereas g is an onto function. Functions which are both oneone and onto are of special importance in mathematics. Let us see what makes them special. Consider a function f: Since f is an onto function, each y E Y is the image of some x E X.
Also, since f is one-one, y cannot be the image of w o distinct members of X. Consequently, f sets up a one-to-one correspondence between the members of X and Y. It is this one-to-one correspondence between members of X and Y which makes a oneone and onto function so special, as we shall soon see.
Consider the function f: We can see that f is one-one as well as onto. The function g so defined is called an inverse off. For this reason g is called the inverse off. As you will notice, the function g is also one-one and onto and therefore it will also have an inverse.
You must have already guessed that the inverse of g is the function f. From this discussion we have the following: Iff is one-one and onro function from X to Y, then there exists a unique function g: The function g so defined is called the inverse of f.
Further, if g is the inverse off, then f is the inv'erse of g, and the two function f and g are said to be the inverses of each other. The inverse of a function f is usually denoted by fI. To find the inverse of a given function f, we proceed as follows: The resulting expressioil for x in terms of y defines the Inverse function.
There is an interesting relation between the graphs of a pair of inverse functions because of which, if the graph of one of them is known, the graph of the other can be obtained easily. Let f: A point p, q lies on the graph o f f c q. Now the points p, q and q, p are reflections of each other with respect to w. Therefore, we can say that the graphs o f f and g are reflections of each other w. Therefore, it follow that, if the graph of one of the functions f and g is given, that of the other can be obtained by reflecting it w.
E E 7 Compare the graphs of In x and ex given in Figs. If a given function is not one-one on its domain, we can choose a subset of the domain on which it is one-one, and then define its inverse function.
For example, consider the function f: In this section we shall see how we can construct new functions from some given functions. We give a few such ways here. In the above example there is nothing special about the number 2. We could have taken any real number to construct a new function from f.
Also, there is nothing special about the particular function that we have considered. We could as well have taken any other function. This suggests the following definition: Let f be a function with domain D and let k be any real number. The scalar nlultiple o f f by k is a function with domain D. Two special cases of the above definition are important.
That is, 0. If k -1, the function kf is called the negative o f f and is denoted simply by -f instead of the clumsy If. Ahsolute Value Function or modulus function of a given function Let f be a. The absolute value function o f f , denoted by I f I and r e d as od f is defined by setting. X I,for all x c D. Therefore, the graphs o f f and f are reflections of each other w.
As an example, consider the graph in Fig. Sum, difference,Product and Quotient of two functions If we are given two functions with a common domain, we can form-several new functions by applying the four fundamental operations of addition, subtraction, multiplication and division on them.
The function d is the function obtained by subtracting g from f, and is denoted by f-g. The function p, called the product of the functions f and g, is denoted by fg. Thus, for all XE D. The function q is called the quotient o f f by g and is denoted by flg.
Remark 5: Example 4: Consider the functions f: All the operations defined on functions till now, were similar to the corresponding operations on real numbers. In the next subsection we are going to introduce an operation which has no parallel in R. Composite functions play a very important role in calculus. You will realise this as you read this course further. Uptill now we have considered functions with the same domain.
We shall now consider a pair of functions such that the co-domain of one is the domain of the other. We define a function h: To obtain h x , we first take the f-image, f x , of an element x of X. We then take the g image of f x , that is, g f x , which is an element of Z. This scheme has been shown in Fig.
The function h, defined above, is called the composite o f f and g and is written as gof. Note the order. We first find thesf-image and then its g-image. Try to distinguish it form fog, which will be defined only when Z is a subset of X. Also, in that case, fog is a function from y to y. Example 5: Thus gofand fogare both defined, but are different from each other. The concept of composite function is used not only to combine functions, but also to look upon a give11function as made up of two simpler functions.
For example, consider the function. Now let us try to find the composites fog and g f of the functions: Or, in other words, each of gof and fogis the identity function on R.
What we have observed here is true for any two functions f and g which are inverses of each other. Thus, iff: Since the domain of gof is X and that of fogis Y, we can write this as: This fact is often used to test whether two given functions are inverses of each other. In this section we shall talk about various types of functions, namely, even, odd, increasing, decreasing and periodic functions. In each case we shall also try to explain the concept through graphs.
Consider the functions f defined on R by setting. This is an example ofan even functi0. We find that the graph a parabola is symmetrical about the y-axis. Such functions are called even functions. The graph of an even function is symmetric with respect to the y-axis. We also note that if the. Thus, if we are required to'draw the graph of an even function, we can use this property,to our advantage.
We only need to draw that part of the graph which lies to the right of the y-axis and then just take its reflection w. The graph of g is shown alongside. The functions f and g above are similar in one respect: Such functions are called odd functions. If x, Qx is a point on the graph of an odd function f, then -x, -f x is also a point on it.
This can be exptessed by saying that the graph of an odd function is symmetric with respect to the origin. In other words, if you turn the graph of an odd function through " about the origin you will find that you get the original graph again. Conversely, if the graph of a function is symmetric with respect to the origin, the function must be an odd function.
The above facts are often usefui while handling odd functions. E E 13 We ate giving below two functions alongwith their graphs.
By calculations as well as by looking at the graphs, find out for each whether it is even or odd. While many of the functions that you will come across in this course will turn out to be either even or odd, there will be many more which will be neither even nor odd. Consider, for example, the function f: The answer is 'no'. Therefore, f is not an even function. Therefore f is not an odd function.
The same conclusion coyld have been drawn by considering the graph o f f which is given m Fig. You will observe that the graph is symmetric neither with respect to the y-axis, nor with respect to the origin. Now there should be no difficulty in solving the exercise below.
E E 14 Which of the following functions are even, which are odd, and which are neither even nor odd? Does the of a company increase with production'! Does the volume of gas decrease with increase in pressure? Problems like these require the use of increasing or decreasing functions.
Now let us see what we mean by an increasing function. Conslder the hnction g and h defined by -x,. In other words, as x increases. This fact can also be seen from the graph of g shown in Fig.
Let us find out how h x behaves as x increases. You can verify this by choosing any values for x, and x,. Equivalently, we can say that h x increases or does not decrease as x increases. The same can be seen from the graph of h in Fig.
Functions like g and h above are called increasing or non-decreasing functions. Clearly, the function g: We shall now study another concept which is, in some sense, complementary to that of an. From the graph we can easily see that as x increases f, does not increase. The graph of f, is shown in Fig. Functions like f, an; f2 are called decreasing or non-increasing functions. The above two examples suggest the follwing definition: A function f defined on a domain D is said to be a monotone function if it is either increasing or decreasing on D.
All the four functions g, h, f,, f, discussed above are monotone functions. The phrases 'monotorlically increasing' and 'monotonically decreasing' are often used for 'increasing' and 'decreasing', respectively. While many functions are monotone, there are many others which are not monotone Consider, for example, the function.
If we find that a given function is not monotone, we can still determine some subsets of the domain on which the function is increasing or decreasing.
E E 15 Given below are the graphs of some funchons. Classify them as non-decreasing. Periodic functions occur very frequently in application of mathematics to various branches of science.
Many phenomena in nature such as propagation of water waves, sound waves, light waves, electromagnetic waves etc. Similarly, weather conditions and prices can also be described in terms of periodic functions. Look at the following patterns:. You must have come across patterns similar to the ones shown in Fig. In each of these patterns a design keeps on repeating itself.
A similar situation occurs in the graphs of periodic functions. Look at the graphs in Fig. In each of the figures shown above the graph consists of a certain pattern repeated infir!
Both these graphs represent periodic functions. To understand the situation, let us examine these graphs closely. Consider the graph in Fig.
This portion is being repeated both to the left as well as to the right. That is to say, if x is any point of [-1, I], then the ordinates at x, x f2, x k 4, x 6, The graph therefore represents the function f defined by. The graph in Fig. You wili notice that the portion of the graph between 0 and 2n is repeated both to the right and to hi: We now give a precise meaning to the tcrnl "a periodic function".
The number p is said to be a pcricd off. The smallest positive ineger p with the property described above is callcd the period off. This means that nn, n E N are all periods of the tangent function. The smallest of nn, that is n, is the period of the tangent function.
See if you can do this exercise. The graph of this function is as shown in Fig. The given function is therefore periodic, the numbers 1,2,3,4 being all periods.
The smallest of these, namely 1, is the period. Thus the given function is periodic and has the period 1. Remark 6 Monotonicity and periodicity are two properties of functions which cannot coexist. A monotoile function can never be periodic, and a periodic function can never be monotone. In general, it may not be easy to decide whether a given function is periodic or not. But sometimes it can be done in a straight forward manner. Suppose we want to find whether the function f: We start by assuming that it is periodic with period p: This is a contradiction.
E E 17 Examine whether the following functions are periodic or not. Write the periods of the periodic functions. Successfully reported this slideshow. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime. Upcoming SlideShare.
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