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 : let x 1a;b, a x 2a;b .

The dependence of the variable y on the variable x, in which each value of x corresponds to a single value of y is called a function. The notation is y=f(x). Each function has a number of basic properties, such as monotonicity, parity, periodicity, and others.

Consider the parity property in more detail.

A function y=f(x) is called even if it satisfies the following two conditions:

2. The value of the function at the point x belonging to the scope of the function must be equal to the value of the function at the point -x. That is, for any point x, from the domain of the function, the following equality f (x) \u003d f (-x) must be true.

Graph of an even function

If you build a graph of an even function, it will be symmetrical about the y-axis.

For example, the function y=x^2 is even. Let's check it out. The domain of definition is the entire numerical axis, which means that it is symmetrical about the point O.

Take an arbitrary x=3. f(x)=3^2=9.

f(-x)=(-3)^2=9. Therefore, f(x) = f(-x). Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of the function y=x^2.

The figure shows that the graph is symmetrical about the y-axis.

Graph of an odd function

A function y=f(x) is called odd if it satisfies the following two conditions:

1. The domain of the given function must be symmetrical about the point O. That is, if some point a belongs to the domain of the function, then the corresponding point -a must also belong to the domain of the given function.

2. For any point x, from the domain of the function, the following equality f (x) \u003d -f (x) must be satisfied.

The graph of an odd function is symmetrical with respect to the point O - the origin. For example, the function y=x^3 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 O.

Take an arbitrary x=2. f(x)=2^3=8.

f(-x)=(-2)^3=-8. Therefore f(x) = -f(x). Thus, both conditions are satisfied for us, which means that the function is odd. Below is a graph of the function y=x^3.

The figure clearly shows that the odd function y=x^3 is symmetrical with respect to the origin.