Prepare a report on the works of M. Lomonosov as a linguist. What is the significance of its linguistic heritage in determining the status of the Russian language? Finding the range of values of a function from its graph
Meaning is one of the most controversial and controversial issues in language theory. The issue of determining the meaning of a word (meaning lexical meaning) is widely covered in the works of domestic and foreign linguists. However, despite its centuries-old history, it has still not received not only a generally accepted, but even a sufficiently clear answer.
In modern linguistics, two approaches to the problem of determining meaning can be distinguished: referential(referential) and functional(functional). Scientists who adhere to the referential approach strive to describe meaning as a component of a word with the help of which a concept is conveyed, and which thus gives the word the ability to objectively reflect existing reality, designate objects, qualities, actions and abstract concepts. Proponents of the functional approach study the functions of words in speech and pay less attention to the question “what is meaning?” than to “what functions does meaning?” Ginzburg R.Z., Khidekel S.S., Kyazeva G.Yu. Lexicology in English. - M., 1979 (in Engl.) - P. 13..
All the major works on the theory of semantics have so far been based on the referential approach. The central idea of this approach is to identify three factors that characterize the meaning of a word: “the word (the symbol)” (sound form of the word), “the mental content” (concept) and “the referent” (the term “referent” - that object (action) , quality), which means the word). In accordance with this approach, meaning is understood as a complex whole, consisting of a designated object and a concept about this object. This relationship is presented by scientists in the form of a schematic image, namely triangles that differ slightly from each other. The most famous is the Ogden-Richards triangle. Stern G. Meaning and change of meaning with special reference to the English language. - Goeteborg, 1931, - P.45., given in the book of the German linguist Gustav Stern “Meaning and change of meaning with special reference to the English language”.
Thought or reference
(the mental content)
Symbol Referent
The term “symbol” here refers to the word; “thought” or “reference” is a concept. The dotted line means that there is no direct connection between the referent and the word: it is established only with the help of the concept. German linguist Gustav Stern argues that the meaning of a word is completely determined by its relationship with its three factors: word, referent and concept. In accordance with the above, G. Stern offers the following definition of the meaning of a word: “The meaning of a word in real speech is identical with those elements of the subjective understanding of the object denoted by the word by the speaker or listener, which, in their opinion, are expressed by this word” Stern G. Ibid. - P. 37..
S. Ulmann Ulmann S. Words and their use. - L., 1951. - P. 32-33., defining the meaning, proposes to simplify the terminology and replaces “symbol” with “name” (name), and “thought or reference” with “sense” (meaning). He also proposes to exclude the term “referent” from the definition, explaining this by the lack of a direct connection between the word and the referent and trying to explain in more detail the connection between the two key terms - name and meaning. The scientist emphasizes the two-way connection between a word and the concept that this word denotes. Not only does a word spoken or written bring to mind the corresponding concept, but the very concept that comes to mind forces us to find the appropriate word. When I think of a table, I will definitely name the word “table”, and also, when I hear the word “table”, I will definitely imagine it. Thus, Ullman arrives at this definition of meaning: Meaning is a reciprocal relationship between the name and the sense, which enables the one to call up the other.(Meaning is a two-way relationship between name and meaning (word and concept), which, when mentioned, allows the first to instantly recall the second and vice versa).
A.I. Smirnitsky Smirnitsky A.I. Lexicology of the English language. - M., 1956. - P. 149-152. states that the meaning of a word cannot be identified with either its referent, i.e. the object that it denotes, nor with the sound of this word. Taking into account the above, he offers the following definition of the meaning of a word: the meaning of a word is this is a known representation of an object, phenomenon or relationship in consciousness(or a mental formation similar in nature, constructed from reflections individual elements reality - mermaid, goblin, witch, etc.), included in the structure of a word as its so-called internal side in relation to which the sound of the word acts as a material shell, necessary not only for expressing meaning and for communicating it to other people, but also for its very emergence, formation, existence and development.
In contrast to proper names, pronouns do not name anything, but only point to someone or something, mainly revealing its relationship to the speaker: you, my, that, hers. The meaning of pronouns is extremely general.
Interjections do not name or indicate anything. Their meaning lies in the fact that they express, not concepts, but the feelings and will of the speaker. An interjection can express a feeling in general: Oh! Ah! Dear me! oh my! oh dear! Or some specific feeling, for example: despondency (alas!), annoyance (damn!), approval (hear! hear!), disdain (pooh!), surprise (gosh!), etc. Imperative, i.e. expressing will, interjections can be a call to calm down or shut up: come, come! easy! there, there! hush! etc.
Although they have meaning, proper names, pronouns and interjections do not express concepts.
The meaning is determined not only by the connection of the word with objects of reality, but also by the place of the word in the system of this language. (Comparing lexical systems different languages, we see that the nature and essence of the dependence of meaning on the structure of language becomes especially obvious). Based on this, the meaning of a word can be defined as the mental content assigned to a given sound form, conditioned by the system of a given language, and common to a given language community. Arbekova T.I. Lexicology of the English language: Practical. well. - M., 1977. - P. 52-53.
The meaning of words is determined by the entire lexical-semantic system of the language and is the result of a reflection of socially conscious objective reality. Lexical meaning is formed under the conditions of specific connections and relationships between words of a given language. Unlike concepts that are common to different languages, the lexical meaning of a word is always nationally specific, like all vocabulary in general. In addition to the concept it expresses, the meaning of a word may also include other components: emotional connotation, stylistic characteristics, correlation with other words of the same language. It is layered with additional ideas and various kinds of semantic associations. Depending on which part of speech a word belongs to, its lexical meaning is associated with a certain range of grammatical meanings and may be influenced by them, so that each part of speech has its own semantic features. The non-identity of meaning and concept is also manifested in the fact that one concept can be expressed by the meaning of two or more words, and, conversely, one polysemantic word can, in its meanings, unite a whole group of interconnected concepts. The lexical meaning of a word may, finally, not coincide with the concept in scope or content. Arnold I.V. Decree. op. - P. 55.
Function y=f(x) is such a dependence of the variable y on the variable x, when each valid value of the variable x corresponds to a single value of the variable y.
Function definition domain D(f) is the set of all acceptable values variable x.
Function Range E(f) is the set of all admissible values of the variable y.
Graph of a function y=f(x) is a set of points on the plane whose coordinates satisfy a given functional dependence, that is, points of the form M (x; f(x)). The graph of a function is a certain line on a plane.
If b=0 , then the function will take the form y=kx and will be called direct proportionality.
D(f) : x \in R;\enspace E(f) : y \in R
Schedule linear function- straight.
The slope k of the straight line y=kx+b is calculated using the following formula:
k= tan \alpha, where \alpha is the angle of inclination of the straight line to the positive direction of the Ox axis.
1) The function increases monotonically for k > 0.
For example: y=x+1
2) The function decreases monotonically as k< 0 .
For example: y=-x+1
3) If k=0, then giving b arbitrary values, we obtain a family of straight lines parallel to the Ox axis.
For example: y=-1
Inverse proportionality
Inverse proportionality called a function of the form y=\frac (k)(x), where k is a non-zero real number
D(f) : x \in \left \( R/x \neq 0 \right \); \: E(f) : y \in \left \(R/y \neq 0 \right \).
Function graph y=\frac (k)(x) is a hyperbole.
1) If k > 0, then the graph of the function will be located in the first and third quarters of the coordinate plane.
For example: y=\frac(1)(x)
2) If k< 0 , то график функции будет располагаться во второй и четвертой координатной плоскости.
For example: y=-\frac(1)(x)
Power function
Power function is a function of the form y=x^n, where n is a non-zero real number
1) If n=2, then y=x^2. D(f) : x \in R; \: E(f) : y \in; main period of the function T=2 \pi
Each function has two variables - an independent variable and a dependent variable, the values of which depend on the values of the independent variable. For example, in the function y = f(x) = 2x + y The independent variable is "x" and the dependent variable is "y" (in other words, "y" is a function of "x"). The valid values of the independent variable "x" are called the domain of the function, and the valid values of the dependent variable "y" are called the domain of the function.
Steps
Part 1
Finding the Domain of a Function-
Select the appropriate entry for the function's scope. The scope of definition is written in square and/or parentheses. Square bracket applies when the value is within the scope of the function; if the value is not within the scope of the definition, a parenthesis is used. If a function has several non-adjacent domains, a “U” symbol is placed between them.
- For example, the scope of [-2,10)U(10,2] includes the values -2 and 2, but does not include the value 10.
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Plot a graph quadratic function. The graph of such a function is a parabola, the branches of which are directed either up or down. Since the parabola increases or decreases along the entire X-axis, the domain of definition of the quadratic function is all real numbers. In other words, the domain of such a function is the set R (R stands for all real numbers).
- To better understand the concept of a function, select any value of “x”, substitute it into the function and find the value of “y”. A pair of values “x” and “y” represent a point with coordinates (x,y) that lies on the graph of the function.
- Plot this point on the coordinate plane and do the same process with a different x value.
- By plotting several points on the coordinate plane, you get general idea about the form of the graph of a function.
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If the function contains a fraction, set its denominator to zero. Remember that you cannot divide by zero. Therefore, by setting the denominator to zero, you will find values of "x" that are not within the domain of the function.
- For example, find the domain of the function f(x) = (x + 1) / (x - 1) .
- Here the denominator is: (x - 1).
- Equate the denominator to zero and find “x”: x - 1 = 0; x = 1.
- Write down the domain of definition of the function. The domain of definition does not include 1, that is, it includes all real numbers except 1. Thus, the domain of definition of the function is: (-∞,1) U (1,∞).
- The notation (-∞,1) U (1,∞) reads like this: the set of all real numbers except 1. The infinity symbol ∞ stands for all real numbers. In our example, all real numbers that are greater than 1 and less than 1 are included in the domain.
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If a function contains a square root, then the radical expression must be greater than or equal to zero. Remember that the square root of negative numbers not extracted. Therefore, any value of “x” at which the radical expression becomes negative must be excluded from the domain of definition of the function.
- For example, find the domain of the function f(x) = √(x + 3).
- Radical expression: (x + 3).
- The radical expression must be greater than or equal to zero: (x + 3) ≥ 0.
- Find "x": x ≥ -3.
- The domain of this function includes the set of all real numbers that are greater than or equal to -3. Thus, the domain of definition is [-3,∞).
Part 2
Finding the range of a quadratic function-
Make sure you are given a quadratic function. The quadratic function has the form: ax 2 + bx + c: f(x) = 2x 2 + 3x + 4. The graph of such a function is a parabola, the branches of which are directed either up or down. Exist various methods finding the range of values of a quadratic function.
- The easiest way to find the range of a function containing a root or fraction is to graph the function using a graphing calculator.
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Find the x coordinate of the vertex of the function graph. For a quadratic function, find the x coordinate of the vertex of the parabola. Remember that the quadratic function is: ax 2 + bx + c. To calculate the x coordinate, use the following equation: x = -b/2a. This equation is a derivative of the basic quadratic function and describes a tangent whose slope is zero (the tangent to the vertex of the parabola is parallel to the X-axis).
- For example, find the range of the function 3x 2 + 6x -2.
- Calculate the x coordinate of the vertex of the parabola: x = -b/2a = -6/(2*3) = -1
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Find the y-coordinate of the vertex of the function graph. To do this, substitute the found “x” coordinate into the function. The desired coordinate “y” represents the limiting value of the function range.
- Calculate the y coordinate: y = 3x 2 + 6x – 2 = 3(-1) 2 + 6(-1) -2 = -5
- The coordinates of the vertex of the parabola of this function are (-1,-5).
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Determine the direction of the parabola by substituting in the function by at least one value "x". Choose any other x value and plug it into the function to calculate the corresponding y value. If the found “y” value is greater than the “y” coordinate of the vertex of the parabola, then the parabola is directed upward. If the found “y” value is less than the “y” coordinate of the vertex of the parabola, then the parabola is directed downward.
- Substitute into the function x = -2: y = 3x 2 + 6x – 2 = y = 3(-2) 2 + 6(-2) – 2 = 12 -12 -2 = -2.
- Coordinates of a point lying on the parabola: (-2,-2).
- The found coordinates indicate that the branches of the parabola are directed upward. Thus, the range of the function includes all values of "y" that are greater than or equal to -5.
- Range of values of this function: [-5, ∞)
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The domain of a function is written similarly to the domain of a function. The square bracket is used when the value is within the range of the function; if the value is not in the range, a parenthesis is used. If a function has several non-adjacent ranges of values, a “U” symbol is placed between them.
- For example, the range [-2,10)U(10,2] includes the values -2 and 2, but does not include the value 10.
- With the infinity symbol ∞, parentheses are always used.
Determine the type of function given to you. The range of values of the function is all valid “x” values (laid along the horizontal axis), which correspond to valid “y” values. The function can be quadratic or contain fractions or roots. To find the domain of a function, you first need to determine the type of the function.
A function is a model. Let's define X as a set of values of an independent variable // independent means any.
A function is a rule with the help of which, for each value of an independent variable from the set X, one can find a unique value of the dependent variable. // i.e. for every x there is one y.
From the definition it follows that there are two concepts - an independent variable (which we denote by x and it can take any value) and a dependent variable (which we denote by y or f (x) and it is calculated from the function when we substitute x).
FOR EXAMPLE y=5+x
1. Independent is x, which means we take any value, let x=3
2. Now let’s calculate y, which means y=5+x=5+3=8. (y depends on x, because whatever x we substitute, we get the same y)
The variable y is said to functionally depend on the variable x and is denoted as follows: y = f (x).
FOR EXAMPLE.
1.y=1/x. (called hyperbole)
2. y=x^2. (called parabola)
3.y=3x+7. (called straight line)
4. y= √ x. (called parabola branch)
The independent variable (which we denote by x) is called the function argument.
Function Domain
The set of all values that a function argument takes is called the function's domain and is denoted D(f) or D(y).
Consider D(y) for 1.,2.,3.,4.
1. D (y)= (∞; 0) and (0;+∞) //the entire set of real numbers except zero.
2. D (y)= (∞; +∞)//all number of real numbers
3. D (y)= (∞; +∞)//all number of real numbers
4. D (y)= )