Математика. Материалы курса

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Operations on functions. The properties of the function.

Today we continue to study the function, and we are going to consider operations on functions and properties of functions.

The first operation is not new for you; it is addition of a function. You need to pay attention to the following in these definitions: they all contain two points.

The first point relates to the domain, and the second one to how the function values are calculated. The sum of a function is defined at the intersection of the domains of two functions, and the values at any point are defined as the sum of the values of these functions on the domain (see the video).

As for other arithmetic operations, the difference and products of functions are defined similarly: these operations are defined at the intersection of the domains, and the value of the function is defined according to the operation (see the video). We define the quotient in a different way. The quotient of the functions f and g is defined on the intersection of the domains of the functions f and g. In addition, we take the points of intersection at which the denominator of the function g is not zero (see video). We define the second point as the quotient of the function values.

Operations on the non-arithmetic function.

The composition of a function f and g is a function that we denote by the symbol °. Besides, it is also called a complex function formed using the function f and g. Where is its domain? It is defined at those points in the domain of the function f where the value of the function f (x) belongs to the domain of the function g. The value of this function is defined as g(f(x)) at points in the specified domain.

Let's consider a simple example. We have the functions f(x) and g(x). Let's make compositions, we read these characters from right to left f and g, and the second composition of the functions g and f. In accordance with the second point, we find the composition of the functions f and g as g(f(x)). We replace f(x) with ln x and calculate the value of the function g. Note the composition function of the functions g and f. As a result, we get a completely different function. Therefore, this example shows that the composition of a function is not a commutative operation. If you change the order of actions, you get a different function.

Another function that is not arithmetic is function narrowing, which results in a new function. What is an E set? This is a subset of the domain. This becomes the domain of the new function. At all points, the values of the new function are found similarly to those of the function f. How is the function narrowing related to the function f? The graph of this function is part of the graph of the function f.

Let's consider a simple example. We have a quadratic function f(x)= х². The graph is a parabola. Let us consider a subset of the domain of the set R. For example, [0;+∞) and we are going to consider the narrowing of the function f in this set. We get a new function because it has a different domain, but the graph of this function is part of the graph of the function f. Note that there is another term: the function f is a continuation regarding the function g.

The next point of our lecture is properties of functions.

Monotonicity. Monotonicity is related to increasing and decreasing of the function. There are quite a few definitions. Let's consider the first of them.

The function is called strictly decreasing on the set E, which is part of the domain, if for any points x₁ and x₂ from the set E satisfying the inequation x₁<x₂, the inequation f (x₁)> f (x₂) is met. How do we writ down the logical form of this definition? This is what you should pay attention to and learn the logical form of notation. What we have just said and what is written here in text, has the following logical form (see the video). We are not going to write the rest of the definitions in words.

Let's consider the general structure of these definitions and the differences.

The following definitions are associated with increasing/decreasing on a set: strict decreasing, decreasing, strict increasing, increasing, and constancy of a function on a set.

If we compare these definitions, we see they are similar except for one point: the last inequations are different, and they reflect the term that is defined. We therefore pronounce the same words that were used before for strict decreasing, with the exception of the last inequation.

What do we call a monotone function? Monotone means both increasing and decreasing. Strictly monotone means either strictly increasing or strictly decreasing. We were talking about the set E. Sometimes the set is not specified, for example, the function is called decreasing (the set is not specified). This means that we are talking about the entire domain. In this case, the definition is as follows: for any x₁ and x₂ from the domain of the function f, if x₁<x₂, the inequation f (x₁) ≥ f (x₂) is satisfied. Similarly, we write the definition of monotonicity for all other cases.

Let us consider an example. The function is set graphically. This is the task for self-study. Name the interval of monotonicity of this function, using the terms that we have just introduced. We are not going to analyze this example, but try to complete it yourself. The next slide shows the answers. You can compare your conclusions with this answer.

Parity, not parity. The definition again contains two points. The function is called even if: for any x, if it belongs to the domain there is minus x also belonging to the domain. The domain of the function is symmetric with respect to zero of the set. And at any point of the domain the value of the function is equal to the value of the function at the symmetric point.

The definition of the odd function is formulated similarly, with the exception of point 2. The domain of definition is symmetric with respect to zero and at symmetric points the values of the function are opposite in sign.

How does the graph show even and odd functions? According to the definition if M (x;y) is a point of the graph, i.e. f(x)=y, and the function is even, then the point N(-x;y) is also a point of the graph. Thus, the graph of the even function is symmetric regarding the Oy axis. For the odd function, if M (x;y) is a point of the graph, then the definition implies that the point K (- x;-y) is also a point of the graph; it is symmetric with respect to the origin coordinate point. This means that the graph of the odd function has the property of central symmetry relative to the origin coordinate point.

Limitation. The function f is bounded from above (on the set E) if there exists a real number b such that for all points x from the function domain, the set E in particular, the inequation f(x) ≤ b is satisfied. You can see the symbolic form of the notation in the video. If the set E is not mentioned, this is the first case. If we speak about the function bounded from above on the set E, then we write for any x from the set E, everything else is the same.

For the lower bound, the situation is similar. First, let's see what it looks like. The function is bounded from above. This means that there is a horizontal line y=b, above which there are no points of the graph of the function f.

For the function bounded from below, we write the definition in the same way (see the video). From the geometrical point of view, bounded from below means the existence of a horizontal line below which there are no points in the graph of the function f.

The definition of the bounded function. Bounded means limited at the top and bottom. To work with the definition, it is convenient for us to use the logical form of notation.

There are two equivalent forms of writing this definition. The function is bounded if there are two numbers a and b, such that the value of the function is between these numbers, whatever the point from the domain. Point 3 in the video shows the definition of boundedness where only one non-negative number m is used.

You can try to prove the equivalence of these three definitions. The boundedness of the function means that there are two horizontal lines and the function graph is between them. That is, the function graph is located in a certain horizontal area.

Periodicity. What is a period? The number T≠0 is the period of the function if two points are met: 1) if a point belongs to the domain, then it contains the points that are shifted by T to the right or left, and they must also belong to the domain, 2) at these points, the function values are equal. The definition of the periodic function: the function is periodic if it has such a period. The definition leads us to the conclusion that, if T is the period of the function, then the period is any number of nT, where n is an integer that is not equal to zero. Therefore, the periodic function has infinitely many periods, including both positive and negative ones. The smallest positive period is the main period, and we denote it by Т₀.

Not every periodic function has the main period. For example, the constant f(x)=1 is a periodic function, its domain is R. All its values are equal at all points, so any number T≠0 is a period, but among the positive numbers there is no smallest number, there is no smallest positive number, so there is no main period. We know that all trigonometric functions: sine, cosine, tangent, and cotangent have a period, and they have a basic period.

Here is the task for you: what is the main period of these functions.

Significant properties of the periodic function that allow us to find out whether a function is periodic or not. What is it related to? If the function takes a value, it takes the latter infinitely many times. Therefore, the following statement is true: if the function f is periodic, then any equation of the form f (x)= a, where a is a certain number, either has no solutions, or has infinitely many solutions.

Let's consider this property in detail. We have f(x)= x + 1. How can we prove that the function is not periodic? Let us consider the equation f(x) = 1. This equation has one solution: x=1. Therefore, the function cannot be periodic. The periodic function property is not met.

Reversibility. Inverse function.

Let the function f be injective. In this case, it is not only injective; it maps the domain of the definition of the set X to the domain of the values of the set Y. Thus, f is a bijection of x on y. In this case, we can change the direction of the arrows, and we get a new function, which we denote f^-1 and call the inverse function. It is defined on the set Y and takes a value in the set X. Look at the drawing, the function f is a mapping from X to Y, and f^-1 is a mapping from Y to X. We have changed the direction of the arrows. Definition: the function f^-1 is called inverse for the function f, 1) if its domain is the set of values of the function f, 2) the value of the function f^-1 (x)=y if, and only if, f(y) = x. The function is invertible if it has an inverse function. The function can also be irreversible.

When is the function invertible? Based on our first arguments, we understand that a function is invertible if and only if it is injective. This is the first statement. We check the function for injectivity: if it is injective, then it is invertible.

Statement Two is based on the properties of a strictly monotone function. If the function is strictly monotone, then it is injective, and therefore invertible.

A property that binds the graphs of the function f and the inverse function. They are symmetric with respect to the bisector 1 and 3 of the coordinate angles of the line y = x.

Let's consider the following argument. Let the value of the function at point a be b, the function is invertible, then the value of the inverse function at point b is a. So, if the point (a; b) is a point in the graph of the function f, then the point (b; a) is a point in the graph of the function f^-1. Since these points are symmetric with respect to the line y = x, we have proved this property. We are not going to discuss the algorithm for finding the inverse functions, you can read about it and analyze specific examples. In practical classes, we will analyze the solution of this problem using this algorithm, so it is in front of you (see the video).