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Numeric Functions

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We start the mathematics course with the first section devoted to mathematical analysis. The central concept of mathematics as a science and, in particular, this section is the concept of a function and its properties. We start the lecture with the principal definition of a function.

So, let X and Y be a set of arbitrary elements. A function or map (these concepts are identical for us) is called a law f, according to which every element of the set X corresponds to exactly one element of the set Y. Note that the keywords in this definition are "every" and "exactly one". These symbols are used to indicate the function. The first one is familiar to you.

 The basic concepts related to the function definition

Let the function f be given from the set X to the set Y. In this case, the set X is called the domain of the function f and it is denoted by symbols.

If the map f to the element x represents element y, then y is called the image of the element x, and x is an inverse image (or preimage) of the element y under the map f. The set of all function images is called its set or value domain and it is denoted by the symbols you know.

 If X and Y are numeric sets, that is, they are made up of real numbers, and then the function f is called numeric. We will sometimes omit this adjective; however, mathematical analysis studies these numerical functions.

A special case of a numerical function is a numerical sequence. So, what is a numerical sequence? This is a numerical function and its domain is the set of all natural numbers. We indicate this set with the capital letter N.

The functions f and g are called equal if two conditions are met. First, the areas of definition of these functions coincide, and second, the values of the functions f and g are equal at any point in the area of definition. Note that there are two items in many definitions related to the concept of functions: the first refers to the area of definition, and the second refers to the values of the function.

The graph of a function. The graph of a numerical function is the set of points in the plane (x, y), it is a pair of numbers; it is presented as a point in two-dimensional space, where x, the first element, is the area of definition, and the second component y is the value of the function at the specified point x.

Not every line on the plane is the graph of a function. Let's look at the example. What should be noted? The graph of a function has the most essential property: any vertical line intersects it at no more than one point. Why cannot a circle be considered as the graph of a function? Let's revise the definition. If we draw a vertical line, like this one, we see that x˳ corresponds to two values: y1 and y2. However, there can exist just one value of the function. So, this curve is not the graph of functions.

 A significant concept associated with the function is the type of function or the type of map. It should be mentioned that the term “the type of map” is used not only in the course of mathematical analysis, but in many sections of mathematics.

The first term is injective mapping. So, f of X to Y is a mapping. It is called injective if any two different elements of the set X correspond to different elements of the set Y. The logical notation of this definition looks like this, and the illustration shows that the injective mapping means that no two different points can be preimages of one. The situation when from two different points the arrows connect at one point is impossible. Thus, it means that the mapping is injective.

 Surjective function. The map f of X to Y is surjective if the set Y is the set of values of the function f. What does it mean? No matter which point y of the set Y you take, there is always a preimage in the set X. Thus, y is the value of the function at the point x.

One more concept related to the types of mapping is bijection. We know that the prefix “bi” denotes “two”. Thus, the map f of X to Y is bijective or one-to-one correspondence if it satisfies two conditions - it is injective and surjective.

Let’s take an example to understand these concepts.

The sets X and Y are given by enumeration of elements. The set X consists of three characters – {a, b, c} – three letters, and the set Y is numeric and it consists of three numbers {1, 2, 3}. Let's try to characterize these correspondences. Consider the correspondence f. First, we find out whether this correspondence is a mapping or a function. So, the element a from the set X corresponds to a single element 1. What do we say about the element b? Arrows in and out of both point 2 and point 3. It turns out that two different elements correspond to the element b. This is contrary to the definition of the function. So, this is not a mapping.

Consider the correspondence b. Element a corresponds to a single element 1, element b -to a single element – 3, element c – to a single element - also 3. So, this is a function or mapping. Let's find out what properties this mapping has. Why is this mapping not injective? Because there are two different points – b and c, which correspond to the same value – 3. So, this is not an injective mapping. As for surjectivity, the set Y is not a set of values. Why? The point 2 of the element of the set Y is not the end of any of the arrows, that is, it does not have a preimage. Thus, this correspondence is a map, which is neither injective, nor surjective, well, therefore it is not bijective.

And the last example is to characterize the correspondence h. It is easy to see that this is a mapping: each element of the set X corresponds to a single element of the set Y. Any two different elements of the set X correspond to different elements of the set Y. This mapping is injective and the set Y is a domain of values. Thus, this correspondence is a mapping that is injective, surjective, and bijective.

Let's move on to the next question - how to set the function. Let's start with what it means to "set a function". This means, first, you need to determine where it is set - the function domain. This is the first point. And the second point is to determine how the values at each point are calculated.

So, the first and most common way to set a function is analytical. The term "analytical" means that we define a function using formulas. Usually it looks like this – lower-case y equals to, and on the right there is some analytical expression containing the variable x. "An analytical expression" means that among the operations there is addition, subtraction, multiplication and division, as well as signs of basic, elementary functions. What is a specific feature here? There may be one formula, but the functions may not be equal.

 Let's take an example. In the first example, the function is set by a formula and the domain is not specified at all, but it is absolutely unambiguous. In this case, the domain is the set of elements x in which the function value is defined.

 Here is the term of a domain. That is, it is calculated, located, and exists. It is not difficult to find, you can solve these tasks, that the domain in this case is a closed ray from 2 to plus infinity

Example Two. Take a look - the formula is the same, only x is greater than or equals to 3.

This means that the domain of this function is all points of the numeric line that are greater than or equal to 3. The formula is the same, the domain is different, the functions are not equal. Among the functions defined analytically, we consider those that are set by different formulas at different intervals.

There are even symbols for some of them. For example, such a well - known function as the number sign. It is its name.

The next method is descriptive. This means that we define the set X (the domain) and describe how the values of functions are calculated.

These examples are numerous.

We will analyze the ones that are important to us in the course of mathematical analysis. So, the concept of the integer part of a real number. The integer part of a real number x is the largest integer, we present it with x in square brackets, and it does not exceed the given number x.

According to this definition, the real number x, for which we find the integer part, is between two integers: on the left there is the integer less than or equal to x - this is the integer part of x, and the next integer is to the right of the number x. Note how the value of this function is calculated. We will consider this function for different real numbers: for the number 5.1, it is 5, for the number -3.7, it is not -3, but the number closest on the left is -4, and for an integer, the integer part is equal to the number itself.

The fractional part of a number is defined as the difference between x and the integer part of x. It is easy to see that the fractional part of any real number is greater than or equal to 0 and less than one. The equation is easily obtained from the formula that introduces a fractional part: any real number is the sum of its integer and fractional parts. The fractional part of the same real numbers is calculated here. You can see it here. So, for an illustration, look at the graphs of these functions.

How the function is set. As an integer part of a number. As a fractional part of a number and the graphs. Well, let's also consider the Dirichlet function. We can also say that it is set descriptively. Its values are equal to zero at all irrational points and equal to one at all rational points.

This is a very significant function that serves as a source of a large number of counterexamples in mathematical analysis and allows us to avoid logical errors in our reasoning.

The next method for setting functions is by a table. Probably most often this method occurs in experimental work of physicists and chemists. Well, we also sometimes use this method in our work. The set X consists of four elements{1, 2, 3, 4} - the numbers one, two, three, four and the function values are shown in the bottom line.

The fourth method is a graphical way to set a function. We use this method really often in mathematical analysis to illustrate concepts, to prove theorems, and to add clarity to our reasoning.

The figure shows that, if a curve is set, how the function value is found. The only thing we should keep in mind is that the line that serves as an illustration of the function, the graph of the function, must have the property of a graph: any vertical line must intersect the graph line just at one point.