What does the record y \u003d f (x) mean in mathematics - Knowledge Hypermarket. What is hidden in the F1 designation on the seeds How the size of the aperture affects the quality of the image

>>Math: What does the notation y = f(x) mean in mathematics

What does the entry y \u003d f (x) mean in mathematics

When studying any real process, they usually pay attention to two quantities involved in the process (in more complex processes, not two quantities are involved, but three, four, etc., but we do not consider such processes yet): one of them changes as if on its own, regardless of anything (we denoted such a variable by the letter x), and the other value takes on values ​​that depend on the selected values ​​of the variable x (we denoted such a dependent variable by the letter y). mathematical model the real process is precisely the record in the mathematical language of the dependence of y on x, i.e. relationships between x and y. Recall once again that by now we have studied the following mathematical models: y = b, y = kx, y = kx + m, y = x 2 .

Do these mathematical models have anything in common? There is! Their structure is the same: y = f(x).

This entry should be understood as follows: there is an expression f (x) with a variable x, with the help of which the values ​​​​of the variable y are found.

Mathematicians prefer the notation y = f(x) for a reason. Let, for example, f (x) \u003d x 2, i.e. we are talking about functions y = x 2. Suppose we need to select several values ​​of the argument and the corresponding values ​​of the function. So far we've written like this:

if x \u003d 1, then y \u003d I 2 \u003d 1;
if x \u003d - 3, then y \u003d (- Z) 2 \u003d 9, etc.

If we use the notation f (x) \u003d x 2, then the notation becomes more economical:

f(1) = 1 2 =1;
f(-3) = (-3) 2 = 9.

So, we got acquainted with one more fragment mathematical language: the phrase "the value of the function y \u003d x 2 at the point x \u003d 2 is 4" is written shorter:

"if y \u003d f (x), where f (x) \u003d x 2, then f (2) \u003d 4."

And here is an example of a reverse translation:

If y \u003d f (x), where f (x) \u003d x 2, then f (- 3) \u003d 9. In another way, the value of the function y \u003d x 2 at the point x \u003d - 3 is 9.

EXAMPLE 1. Given a function y \u003d f (x), where f (x) \u003d x 3. Calculate:

a) f(1); b) f(- 4); c) f(o); d) f(2a);
e) f(a-1); f) f(3x); g) f(-x).

Solution. In all cases, the plan of action is the same: in the expression f(x), you need to substitute instead of x the value of the argument that is indicated in brackets, and perform the appropriate calculations and transformations. We have:

Comment. Of course, instead of the letter f, you can use any other letter (mostly from the Latin alphabet): g (x), h (x), s (x), etc.

Example 2 Two functions are given: y \u003d f (x), where f (x) \u003d x 2, and y \u003d g (x), where g (x) \u003d x 3. Prove that:

a) f(-x) = f(x); b) g(-x)=-g(x).

Solution. a) Since f (x) \u003d x 2, then f (- x) \u003d (- x) 2 \u003d x 2. So, f (x) \u003d x 2, f (- x) \u003d x 2, then f (- x) \u003d f (x)

b) Since g (x) \u003d x 3, then g (- x) \u003d -x 3, i.e. g(-x) = -g(x).

The use of a mathematical model of the form y = f(x) turns out to be convenient in many cases, in particular, when the real process is described by different formulas at different intervals of change of the independent variable.

Let us describe some properties of the function y - f (x) using the graph constructed in Figure 68 - such a description of the properties is usually called reading the graph.

Reading a graph is a kind of transition from a geometric model (from a graphical model) to a verbal model (to a description of the properties of a function). BUT
plotting is a transition from an analytical model (it is presented in the condition of example 4) to a geometric model.

So, let's start reading the graph of the function y \u003d f (x) (see Fig. 68).

1. The independent variable x runs through all values ​​from -4 to 4. In other words, for each value of x from the segment [-4, 4], you can calculate the value of the function f(x). They say this: [-4, 4] - the scope of the function.

Why, when solving example 4, did we say that it was impossible to find f(5)? Yes, because the value x = 5 does not belong to the scope of the function.

2. y naim = -2 (the function reaches this value at x = -4); At the nanb. = 2 (the function reaches this value at any point of the half-interval (0, 4].

3. y = 0 if 1 = -2 and if x = 0; at these points, the graph of the function y = f(x) intersects the x-axis.

4. y > 0 if x є (-2, 0) or if x є (0, 4]; at these intervals, the graph of the function y \u003d f (x) is located above the x-axis.

5. y< 0, если же [- 4, - 2); на этом промежутке график функции у = f(x) расположен ниже оси х.

6. The function increases on the interval [-4, -1], decreases on the interval [-1, 0] and is constant (neither increasing nor decreasing) on ​​the half-interval (0,4).

As we study new properties of functions, the process of reading the graph will become more intense, meaningful and interesting.

Let us discuss one of these new properties. The graph of the function considered in example 4 consists of three branches (of three "pieces"). The first and second branches (a straight line segment y \u003d x + 2 and part of the parabola) are “joined” successfully: the segment ends at the point (-1; 1), and the parabola section begins at the same point. But the second and third branches are less successfully “joined”: the third branch (“piece” of the horizontal line) does not begin at the point (0; 0), but at the point (0; 4). Mathematicians say this: "the function y = f(x) undergoes a break at x = 0 (or at the point x = 0)". If the function has no discontinuity points, then it is called continuous. So, all the functions that we met in the previous paragraphs (y = b, y = kx, y = kx + m, y = x2) are continuous.

Example 5. Given a function. It is required to construct and read its schedule.

Solution. As you can see, here the function is given by a rather complicated expression. But mathematics is a single and integral science, its sections are closely related to each other. Let's use what we learned in Chapter 5 and reduce algebraic fraction

is valid only under the restriction Therefore, we can reformulate the problem as follows: instead of the function y = x 2
we will consider the function y \u003d x 2, where We construct a parabola y \u003d x 2 on the coordinate plane xOy.
The line x = 2 intersects it at the point (2; 4). But according to the condition, it means that we must exclude the point (2; 4) of the parabola from consideration, for which we mark this point in the drawing with a light circle.

Thus, the graph of the function is built - it is a parabola y \u003d x 2 with a “punched out” point (2; 4) (Fig. 69).

Let's move on to describing the properties of the function y \u003d f (x), i.e., to reading its graph:

1. The independent variable x takes on any values ​​except x = 2. This means that the domain of the function consists of two open rays (- 0 o, 2) and

2. y max = 0 (achieved at x = 0), y max _ does not exist.

3. The function is not continuous, it undergoes a discontinuity at x = 2 (at the point x = 2).

4. y = 0 if x = 0.

5. y\u003e 0 if x є (-oo, 0), if x є (0, 2) and if x є (B, + oo).
6. The function decreases on the ray (- ω, 0], increases on the half-interval .

Calendar-thematic planning in mathematics, video in mathematics online, Math at school download

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

Lesson content lesson summary support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-examination workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures graphics, tables, schemes humor, anecdotes, jokes, comics parables, sayings, crossword puzzles, quotes Add-ons abstracts articles chips for inquisitive cheat sheets textbooks basic and additional glossary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in the textbook elements of innovation in the lesson replacing obsolete knowledge with new ones Only for teachers perfect lessons calendar plan for the year methodological recommendations of the discussion program Integrated Lessons

If you look through all the bags of seeds, hung or laid out on the counter, you can often see the designation "F1" indicated somewhere in the corner. This marking is quite common, and you can see it on all types of vegetable crops. So what does F1 mean on seeds? What features and characteristics are embedded in this designation? Let's try to understand this abbreviation.

A little about selection or what F1 means on seeds

With the beginning of the gardening period, or more simply, with the onset of spring, all summer residents think about the issue of planting crops - about what will be planted, where to plant it and in what sequence. But in any case, the garden begins with seeds - be it seeds that are independently collected from grown crops, or purchased in a store, on the market.

Buying seeds is not an easy task, because you need not only to choose the same variety from the variety presented, but also pay attention to the characteristics of the crop. And seeds marked F1 are also usually expensive. And what is their quality? And can you then collect your own seeds from them?

What are F1 varieties and how do they differ from regular seeds

In general, the formula F1 refers to hybrid seeds. It comes from the Italian filli, which means “children”, and the one appears as a symbol of the first generation. That is, hybrids are varieties obtained from crossing two other ordinary varieties of a crop, which are the parents of the variety with the designation F1.

Ordinary varietal seeds are obtained during a long selection process, and carry invariable characteristics, such as yield, fruit color and size, vegetable taste, resistance to diseases, pests, weather conditions, etc. Over time, these characteristics of these varieties do not change. That is, seeds from crops grown from ordinary varietal seed will give exactly the same fruits as their parents.

With hybrid seeds, things are different. They inherit the best qualities from their parents, give themselves completely - they grow quickly and give a 100% abundant and beautiful harvest (with the right agricultural technology). But, unfortunately, their signs are not transmitted, so to speak, “by inheritance”. From seeds from vegetables grown from seed F1, you cannot get exactly the same crops with the same excellent characteristics.

What are the positive characteristics of hybrid seeds?

Breeding hybrid seeds is not accidental. When crossing, they take the best characteristics of their parents that the latter possess. That is, hybrid seeds take away the dominant, pronounced, signs of their parents, and this is what breeders are guided by when breeding a hybrid.

Therefore, as a rule, F1 seeds have an increased yield, resistance to adverse weather conditions, successfully resist diseases and pests, the fruits are large and even, and are characterized by accelerated growth. As a result, they have a hardiness that ordinary varietal seeds do not have. This is why hybrid seeds (assuming they are true hybrid seeds, of course) germinate even when others don't, and produce good yields in what are considered low yield years. In addition, hybrids are most often self-pollinated, and this is a definite plus.

Of course, given these indicators, it is natural that the cost of seeds with the designation F1 differs from ordinary varieties - they are more expensive. Yes, and getting them takes a lot more time and effort. By purchasing a real hybrid, you can be sure that it will give a good harvest (sometimes even in bad weather conditions) on time, or maybe earlier, and its fruits will be large, smooth and fleshy.

How F1 hybrids are made

Hybrid seeds are obtained by crossing varietal seeds. This process is long, moreover, it is done manually, which partly explains the increased cost of the final planting material.

Since F1 seeds obtained by crossing take their dominant traits from their parents, the selection of crossed varieties is carefully considered. For example, they take one variety with increased disease resistance characteristics, and the second variety is abundantly productive. As a result, the producer receives a hybrid that will give a good and healthy mega-crop, and not a single vegetable bush will die from garden diseases.

Or, for example, the first variety will have the main feature of early ripening, and the second - the large size of the fruit, as a result, a large crop will be obtained quickly, even before the ripening period of ordinary varieties. Or one parent will give resistance to bad weather, and the second - precocity. And such signs for each specific species are the sea, and they are transmitted to the seeds of F1 almost in one hundred percent. Although in some cases, "children" outnumber "parents" by 20 percent. Obtaining a unique hybrid is kept secret by producers - from which varieties it was bred.

But obtaining such seeds is troublesome. Firstly, those varieties from which they prefer to get a hybrid are grown in protected ground. But parents must have not only pronounced dominant characters, they must be of the same species, and also be self-pollinating. On one of the plants, at the moment it begins to bloom, the stamens are forcibly removed. And pollen is collected from a plant of a different variety, which, of course, happens with the help of special equipment. The first plant is treated with the obtained pollen. This process lasts for several months, every day, resulting in hybrid seeds.

Disadvantages of F1 seeds

We found out about the excellent qualities and positive aspects of using F1 seeds when growing a crop. But there is a price to pay for all the pleasures in life. So what negative awaits us when using these seeds?

• First, as we said, cost. The price for hybrid seeds exceeds (and sometimes several times) the prices of ordinary varietal ones.
• Secondly, it is impossible to grow a crop from F1 seeds for the next year. As mentioned above, the second generation of hybrid seeds does not inherit the characteristics of their parents - neither yield, nor early maturity, nor dimensional qualities of fruits, nor resistance to diseases and weather. In other words, it is not worth harvesting seeds from vegetables obtained from F1 planting material - this is from the category of “monkey labor”. These harvested seeds of the second generation may not produce what you expect, and they will grow into an incomprehensible variety of infertile crops. Or fruitful, but not with the expected quality.
• Thirdly, if we turn to the biochemical composition of varietal plants and plants grown from F1 seeds, it is different. Adherents of all natural suggest that the first group is closer to wild plants, which means that it is the usual breeding varieties that produce vegetables rich in trace elements and vitamins, while hybrids do not have such a quantity at all. Nonsense, of course - their amino acid composition is normal, but whether the plant has accumulated a sufficient amount of sugars depends on the growing conditions. It is unlikely that a vegetable intended for growing indoors will pick up the "due" glucose in the garden. Therefore, we will single out this point separately.
• Fourth, agricultural technology. Still, no matter what super-properties a hybrid has, it reveals all its excellent characteristics only with proper care. Otherwise, he does not always show them.
• Well, and fifthly, taste. Unfortunately, hybrids do not get all the variety and nuances of taste from their parents. Sometimes they significantly lose to varietal crops in terms of taste, but this is not always the case. Why are hybrid crops thought to taste more like oak? Perhaps this impression was fixed from the purchase of winter greenhouse vegetables. But after all, it is understandable - with a lack of light, photosynthesis is less pronounced, and less glucose is produced.

We can take strawberries as an example - it is clear that wild strawberries are tastier and more aromatic than they are from the garden, and large store-bought strawberries (especially in winter) cannot be compared with them, which has only a small fraction of real taste.

But we, for example, know the most excellent sweet tomatoes from the F1 series - "Red Date", "Yellow Date" and "Orange Date". Our grandchildren love to eat them right from the garden. But last rainy summer, they didn’t pick up sweetness - the taste was almost neutral.

Another thing is that when choosing certain qualities in hybridization, an unsuccessful combination can really turn out. For example, the genes for perfect roundness and the genes for red color can combine to produce absolutely beautiful fruits with no taste. Or in pursuit of a hybrid resistant to late blight, we get a sour hybrid.

That is why we prefer to choose crooked-oblique-yellow-green-orange-black-variegated tomatoes. First, there should be variety in the beds. Secondly, if the weather fails, then the taste of your favorite variety can be replaced by an understudy. Yes, sometimes you want to try something new. But over time, the list of preferences settled down, there is always a gentleman's set of "favorites" for landing.

The nuances of growing bunch cucumbers

I would like to add that the taste of hybrids may not live up to expectations, not only because of crossing, but also because of the flaws in agricultural technology. This is especially clearly seen in hybrids of cucumbers, which give a bundle ovary (10-15 greens are formed in the sinuses). Almost all of our friends bought such F1 varieties and were disappointed - none of them had a picture from the cover. Most likely, the scheme of plant formation is simply not taken into account. And on the bag of seeds there must be a drawing. Briefly, the meaning of the formation is as follows:

• you need to keep the central lash, and not transfer it to lateral shoots, as was customary when growing old varieties;
• blind the lower 5-10 (depending on the variety) of the node - leave only the leaves, and remove the side branches and embryos of the greens completely.

That is, the technique is the same as for peppers, when we remove the first ovary, we “save” strength and nutrients for future abundant fruiting. The plant must develop a good root system and gain what is called the appropriate green mass, then the harvest will be impressive.

And if you do not blind, then the plant produces as usual - one, well, two cucumbers is formed in each node. But they are early, you say. But for the early ones, it is possible to plant cheaper planting material, right? Why ruin the elite rastyukha?

We hope you figured out what the abbreviation F1 means on seeds, and you can safely select varieties for open and closed ground. Do not stop at one variety, grow a wide range of even one crop - this will save you from disappointment in a bad year and there will be something to compare!

Camera aperture - what is it all about? And why is this value indicated after the number of pixels in the smartphone's photomatrix? Do not know? Let's figure it out, along the way, finding out which of the apertures is better.

What is aperture?

Simply put, the aperture is the pupil. Light travels through the cornea (lens), passes through the pupil (aperture/diaphragm), and enters the optic nerve (photomatrix). Why is there an aperture in this chain? Yes, then, to dose the light radiation. The larger it is (the pupil dilates), the more light will hit the matrix (optic nerve).

Aperture f 2.0 - what does it mean? What is aperture measured in?

From the characteristics of smartphones, it is clear that the aperture is measured in special units - f-numbers. Or, as professional photographers say, in f-stops. Moreover, the size range of the aperture consists of fractional numbers - f / 1.4, f / 2.0 and so on. Sometimes a simplified version of the designation is written in the characteristics - aperture 1.8. However, the exact display of this value requires the following spelling - f / 1.8.

According to the laws of mathematics, the maximum value of the aperture is achieved at the minimum value of the divisor - the numerical coefficient located on the right. That is, an aperture of 2.0 (f / 2.0) implies a greater degree of “expansion” of the pupil-diaphragm than an aperture of 2.2 (f / 2.2). And the larger the number on the right, the smaller the degree of aperture opening.

How does aperture size affect image quality?

A large aperture allows the lens shutters to open to the maximum, letting a very large portion of light into the sensor. A small aperture means that the lens shutters are not fully opened, and let a minimum of light into the matrix.

How does this affect image quality? Yes, in the most direct way! A large aperture in bright light is likely to spoil (light up) the frame. Try to take a photo with the sun behind you and you will see all the consequences of too large an aperture. However, another situation is also possible, when a too small aperture value does not allow the matrix to capture a sufficient portion of light and the picture turns out to be dark.

That is, a good aperture can be neither large nor small. It must match the specific shooting conditions. However, in low light conditions, you need the largest possible aperture in order to capture the maximum light. And you shouldn't forget about it.

Is a small aperture really bad?

Not really. At small apertures - from f 4.0 - f 8.0 and below - there is an interesting opportunity to increase the depth of field of the matrix. The smaller the aperture, the more objects are in the focus of the camera. Therefore, small apertures are loved by all fans of landscape photography and portrait photographers who want to get clear pictures without blurring the contours and other noise.

Finally, choosing between aperture f 2.0 and f 2.2 which can't be said better. The first value guarantees the possibility of improving the quality of the photo in a dark room. The second promises to increase the sharpness of the image.

Choosing a smartphone by camera aperture

The trouble with any camera of any smartphone is the very small physical size of the photomatrix (the optic nerve of a mobile device). Therefore, the standard aperture of the main camera is f 2.0 or f 2.2. No smartphone manufacturer that respects its customers will dare to set a smaller aperture value. In this case, the photos in the rooms will be completely unreadable.

The smartphone also does not need a too large value of the f-number. It is easy to oversaturate a small matrix with light, spoiling the balance of the picture. However, devices with a dual camera and an aperture of f / 1.7 have recently appeared, which is very good for a smartphone with an enlarged photomatrix. The quality of pictures in the room of such smartphones is at an unattainable height.

And what is the aperture of the flagships?

At the moment, the champions in the value of f-numbers are the following smartphones:

For the rest, including the vaunted one, the aperture does not exceed f / 2.2.