Chart conversion.
Verbal description of the function.
Graphic way.
The graphical way of specifying a function is the most illustrative and is often used in engineering. In mathematical analysis, the graphical way of specifying functions is used as an illustration.
Function Graph f is the set of all points (x; y) of the coordinate plane, where y=f(x), and x “runs through” the entire domain of the given function.
A subset of the coordinate plane is a graph of some function if it has at most one common point with any line parallel to the Oy axis.
Example. Are the figures below graphs of functions?
The advantage of a graphic task is its clarity. You can immediately see how the function behaves, where it increases, where it decreases. From the graph, you can immediately find out some important characteristics of the function.
In general, analytical and graphical ways of defining a function go hand in hand. Working with the formula helps to build a graph. And the graph often suggests solutions that you won’t notice in the formula.
Almost any student knows the three ways to define a function that we have just covered.
Let's try to answer the question: "Are there other ways to define a function?"
There is such a way.
A function can be quite unambiguously defined in words.
For example, the function y=2x can be defined by the following verbal description: each real value of the argument x is assigned its doubled value. The rule is set, the function is set.
Moreover, it is possible to specify a function verbally, which is extremely difficult, if not impossible, to specify by a formula.
For example: each value of the natural argument x is associated with the sum of the digits that make up the value of x. For example, if x=3, then y=3. If x=257, then y=2+5+7=14. And so on. It is difficult to write this down in a formula. But the table is easy to make.
The method of verbal description is a rather rarely used method. But sometimes it happens.
If there is a law of one-to-one correspondence between x and y, then there is a function. What law, in what form it is expressed - by a formula, tablet, graph, words - does not change the essence of the matter.
Consider functions whose domains of definition are symmetrical with respect to the origin of coordinates, i.e. for anyone X out of scope number (- X) also belongs to the domain of definition. Among these functions are even and odd.
Definition. The function f is called even, if for any X out of its domain
Example. Consider the function 
She is even. Let's check it out.
For anyone X the equalities
Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of this function.
Definition. The function f is called odd, if for any X out of its domain 
Example. Consider the function 
She is odd. Let's check it out.
The domain of definition is the entire numerical axis, which means that it is symmetrical about the point (0; 0).
For anyone X the equalities
Thus, both conditions are satisfied for us, which means that the function is odd. Below is a graph of this function.
The graphs shown in the first and third figures are symmetrical about the y-axis, and the graphs shown in the second and fourth figures are symmetrical about the origin.
Which of the functions whose graphs are shown in the figures are even, and which are odd?
Even function.
Even A function whose sign does not change when the sign is changed is called x.
x equality f(–x) = f(x). Sign x does not affect sign y.
The graph of an even function is symmetrical about the coordinate axis (Fig. 1).
Even function examples:
y= cos x
y = x 2
y = –x 2
y = x 4
y = x 6
y = x 2 + x
Explanation:
Let's take a function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect sign y. The graph is symmetrical about the coordinate axis. This is an even function.
odd function.
odd is a function whose sign changes when the sign is changed x.
In other words, for any value x equality f(–x) = –f(x).
The graph of an odd function is symmetrical with respect to the origin (Fig. 2).
Examples of an odd function:
y= sin x
y = x 3
y = –x 3
Explanation:
Take the function y = - x 3 .
All values at it will have a minus sign. That is the sign x affects the sign y. If the independent variable is a positive number, then the function is positive; if the independent variable is a negative number, then the function is negative: f(–x) = –f(x).
The graph of the function is symmetrical about the origin. This is an odd function.
Properties of even and odd functions:
NOTE:
Not all features are even or odd. There are functions that are not subject to such gradation. For example, the root function at = √X does not apply to either even or odd functions (Fig. 3). When listing the properties of such functions, an appropriate description should be given: neither even nor odd.
Periodic functions.
As you know, periodicity is the repetition of certain processes at a certain interval. The functions describing these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.
Graphs of even and odd functions have the following features:
If a function is even, then its graph is symmetrical about the y-axis. If a function is odd, then its graph is symmetrical about the origin.
Example. Plot the function \(y=\left|x \right|\).Solution. Consider the function: \(f\left(x \right)=\left|x \right|\) and substitute \(x \) for the opposite \(-x \). As a result of simple transformations, we get: $$f\left(-x \right)=\left|-x \right|=\left|x \right|=f\left(x \right)$$ In other words, if replace the argument with the opposite sign, the function will not change.
This means that this function is even, and its graph will be symmetrical about the y-axis (vertical axis). The graph of this function is shown in the figure on the left. This means that when plotting a graph, you can build only half, and the second part (to the left of the vertical axis, draw already symmetrically to the right side). By determining the symmetry of a function before starting to plot its graph, you can greatly simplify the process of constructing or studying a function. If it is difficult to perform a check in a general form, you can do it easier: substitute the same values \u200b\u200bof different signs into the equation. For example -5 and 5. If the values of the function are the same, then we can hope that the function will be even. From a mathematical point of view, this approach is not entirely correct, but from a practical point of view, it is convenient. To increase the reliability of the result, you can substitute several pairs of such opposite values.
Example. Plot the function \(y=x\left|x \right|\).
Solution. Let's check the same as in the previous example: $$f\left(-x \right)=x\left|-x \right|=-x\left|x \right|=-f\left(x \right) $$ This means that the original function is odd (the sign of the function is reversed).
Conclusion: the function is symmetrical with respect to the origin. You can build only one half, and draw the other half symmetrically. This symmetry is more difficult to draw. This means that you are looking at the chart from the other side of the sheet, and even turned upside down. And you can also do this: take the drawn part and rotate it around the origin by 180 degrees counterclockwise.
Example. Plot the function \(y=x^3+x^2\).
Solution. Let's perform the same sign change check as in the previous two examples. $$f\left(-x \right)=\left(-x \right)^3+\left(-x \right)^2=-x^2+x^2$$ $$f\left(-x \right)\not=f\left(x \right),f\left(-x \right)\not=-f\left(x \right)$$ Which means that the function is neither even nor odd.
Conclusion: the function is not symmetrical either about the origin or about the center of the coordinate system. This happened because it is the sum of two functions: even and odd. The same situation will be if you subtract two different functions. But multiplication or division will lead to a different result. For example, the product of an even and an odd function gives an odd one. Or the quotient of two odd leads to an even function.
Even and odd functions are one of its main properties, and parity occupies an impressive part of the school course in mathematics. It largely determines the nature of the behavior of the function and greatly facilitates the construction of the corresponding graph.
Let us define the parity of the function. Generally speaking, the function under study is considered even if for opposite values of the independent variable (x) located in its domain, the corresponding values of y (function) are equal.
Let us give a more rigorous definition. Consider some function f (x), which is defined in the domain D. It will be even if for any point x located in the domain of definition:
- -x (opposite dot) also lies in the given scope,
- f(-x) = f(x).
From the above definition, the condition necessary for the domain of definition of such a function follows, namely, symmetry with respect to the point O, which is the origin of coordinates, since if some point b is contained in the domain of definition of an even function, then the corresponding point - b also lies in this domain. From the foregoing, therefore, the conclusion follows: an even function has a form that is symmetrical with respect to the ordinate axis (Oy).
How to determine the parity of a function in practice?
Let it be given using the formula h(x)=11^x+11^(-x). Following the algorithm that follows directly from the definition, we first of all study its domain of definition. Obviously, it is defined for all values of the argument, that is, the first condition is satisfied.
The next step is to substitute the argument (x) with its opposite value (-x).
We get:
h(-x) = 11^(-x) + 11^x.
Since addition satisfies the commutative (displacement) law, it is obvious that h(-x) = h(x) and the given functional dependence is even.
Let's check the evenness of the function h(x)=11^x-11^(-x). Following the same algorithm, we get h(-x) = 11^(-x) -11^x. Taking out the minus, as a result, we have
h(-x)=-(11^x-11^(-x))=- h(x). Hence h(x) is odd.
By the way, it should be recalled that there are functions that cannot be classified according to these criteria, they are called neither even nor odd.
Even functions have a number of interesting properties:
- as a result of the addition of similar functions, an even one is obtained;
- as a result of subtracting such functions, an even one is obtained;
- even, also even;
- as a result of multiplying two such functions, an even one is obtained;
- as a result of multiplication of odd and even functions, an odd one is obtained;
- as a result of dividing the odd and even functions, an odd one is obtained;
- the derivative of such a function is odd;
- If we square an odd function, we get an even one.
The parity of a function can be used in solving equations.
To solve an equation like g(x) = 0, where the left side of the equation is an even function, it will be quite enough to find its solution for non-negative values of the variable. The obtained roots of the equation must be combined with opposite numbers. One of them is subject to verification.
The same is successfully used to solve non-standard problems with a parameter.
For example, is there any value for the parameter a that would make the equation 2x^6-x^4-ax^2=1 have three roots?
If we take into account that the variable enters the equation in even powers, then it is clear that replacing x with -x will not change the given equation. It follows that if a certain number is its root, then so is the opposite number. The conclusion is obvious: the roots of the equation, other than zero, are included in the set of its solutions in “pairs”.
It is clear that the number 0 itself is not, that is, the number of roots of such an equation can only be even and, naturally, for any value of the parameter it cannot have three roots.
But the number of roots of the equation 2^x+ 2^(-x)=ax^4+2x^2+2 can be odd, and for any value of the parameter. Indeed, it is easy to check that the set of roots of a given equation contains solutions in "pairs". Let's check if 0 is a root. When substituting it into the equation, we get 2=2. Thus, in addition to "paired" 0 is also a root, which proves their odd number.
Which to one degree or another were familiar to you. It was also noted there that the stock of function properties will be gradually replenished. Two new properties will be discussed in this section.
Definition 1.
The function y \u003d f (x), x є X, is called even if for any value x from the set X the equality f (-x) \u003d f (x) is true.
Definition 2.
The function y \u003d f (x), x є X, is called odd if for any value x from the set X the equality f (-x) \u003d -f (x) is true.
Prove that y = x 4 is an even function.
Solution. We have: f (x) \u003d x 4, f (-x) \u003d (-x) 4. But (-x) 4 = x 4 . Hence, for any x, the equality f (-x) = f (x), i.e. the function is even.
Similarly, it can be proved that the functions y - x 2, y \u003d x 6, y - x 8 are even.
Prove that y = x 3 is an odd function.
Solution. We have: f (x) \u003d x 3, f (-x) \u003d (-x) 3. But (-x) 3 = -x 3 . Hence, for any x, the equality f (-x) \u003d -f (x), i.e. the function is odd.
Similarly, it can be proved that the functions y \u003d x, y \u003d x 5, y \u003d x 7 are odd.
You and I have repeatedly convinced ourselves that new terms in mathematics most often have an “earthly” origin, i.e. they can be explained in some way. This is the case for both even and odd functions. See: y - x 3, y \u003d x 5, y \u003d x 7 are odd functions, while y \u003d x 2, y \u003d x 4, y \u003d x 6 are even functions. And in general, for any function of the form y \u003d x "(below we will specifically study these functions), where n is a natural number, we can conclude: if n is an odd number, then the function y \u003d x" is odd; if n is an even number, then the function y = xn is even.
There are also functions that are neither even nor odd. Such, for example, is the function y \u003d 2x + 3. Indeed, f (1) \u003d 5, and f (-1) \u003d 1. As you can see, here Hence, neither the identity f (-x) \u003d f ( x), nor the identity f(-x) = -f(x).
So, a function can be even, odd, or neither.
The study of the question of whether a given function is even or odd is usually called the study of the function for parity.
Definitions 1 and 2 deal with the values of the function at the points x and -x. This assumes that the function is defined both at the point x and at the point -x. This means that the point -x belongs to the domain of the function at the same time as the point x. If a numerical set X together with each of its elements x contains the opposite element -x, then X is called a symmetric set. Let's say (-2, 2), [-5, 5], (-oo, +oo) are symmetric sets, while )
