Parity functions

Even or odd functions are one of its main properties, and the study of the function parity is an impressive part of the school course in mathematics. It is many determines the nature of the behaviour of the function and greatly facilitates building an appropriate graph.

We Define the parity function. Generally speaking, the investigated feature is considered even if for the opposite values of the independent variable (x) within its scope, the corresponding values y (functions) are equal.

Will Give a more rigorous definition. Consider some function f (x) which is given in D. It will be even if for any point x in the region definitions:

• -x (the opposite point) is also in this scope,

• f (-x) = f (x).

This definition indicates the condition necessary for determining such a function, namely, the symmetry relative to the point O being the origin because if a point in b contains region definitions even function, then the corresponding point b also lies in this region. From the foregoing, thus, follows the conclusion: an even function is symmetric with respect to the axis of ordinates (Oy).

In practice, to determine the parity of a function?

Let the functional dependence is defined by the formula h(x)=11^x+11^(-x). Following the algorithm follows directly from the definition, we investigate first of all its scope. It is obvious that it is defined for all values of the argument, that is, the first condition is met.

Recommended

"Knowledge is light and ignorance is darkness": the value, meaning and alternatives

There are some sayings that would seem to need no explanation, such as “teaching & ndash; light and ignorance – darkness”. But some still do not understand their meaning. But not only for such people is written by our article. I...

What was invented by Mendeleev for the army. The history and fate of the invention

D. I. Mendeleev was a brilliant Russian scientist-polymath, who made many important discoveries in various fields of science and technology. Many people know that he is the author of “Fundamentals of chemistry" and the periodic law of chem...

The origin of the Slavs. The influence of different cultures

Slavs (under this name), according to some researchers, appeared in the story only in 6 century ad. However, the language of nationality bears the archaic features of the Indo-European community. This, in turn, suggests that the origin of the Slavs h...

The Next step we substitute instead of the argument (x) to its opposite value (-x).
Receive :
h(-x) = 11^(-x) + 11^x.
Since the addition satisfies the commutative (commutative) law, then obviously, h (x) = h(x) and a given functional dependence – even.

Will Check the parity of the function h(x)=11^x 11^(-x). Following the same algorithm, we obtain that h(-x) = 11^(-x) -11^x. Minus the endure, in the end, have
h(-x)=-( 11^x 11^(-x))=- h(x). Therefore, h(x) – odd.

By the Way, it should be recalled that there are functions that cannot be classified according to these characteristics, they are called neither even nor odd.

Even functions have several interesting properties:

• The result of the addition of such functions, we get an even;
• The subtraction of such functions, we get an even;
• The inverse is even, is also even;
• The result of multiplication of two such functions, we get an even;
• The result of multiplying odd and even functions, we get an odd;
• The result of dividing odd and even functions, we get an odd;
• The derivative of such a function – odd;
• If you build an odd function in the square , get even.

Parity functions can be used in the solution of equations.

To solve the equation g(x) = 0, where the left part of the equation is an even function, would be sufficient to find its solution for the nonnegative values of the variable. Obtained the roots of the equation must be integrated with the opposite numbers. One of them is subject to verification.

This function is successfully used to solve non-standard tasks with the parameter.

For Example, is there any value of the parameter a at which the equation 2x^6-x^4-ax^2=1 will have three roots?

Considering that the variable enters the equation in even degrees, it is clear that replacing x by-x the given equation will not change. It follows that if a number is a root, then it is opposite number. The conclusion is obvious: the roots of the equation different from zero, are included in many it solutions “pairs”.

It is Clear that the number 0 is a root of the equation is not, that is the number of roots of such equations can be only even and, of course, for any parameter value 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 this equation contains solutions “pairs”. Check whether 0 is a root. By substituting it into the equation, we get 2=2 . Thus, in addition to “pair” 0 is also a root, which proves their odd number.