parity and odd functions are one of its main features, and research functions of the parity has an impressive part of the school course in mathematics.It is largely determined by the behavior of functions and greatly facilitates the construction of the corresponding schedule.
define the parity function.Generally speaking, think of the function even if for opposite values of the independent variable (x), under its domain, the corresponding values of y (functions) are equal.
We give a rigorous definition.Consider a function f (x), which is defined in D. It will be even if, for any two points x, located in the domain:
- -x (opposite point) is also in this domain,
- f(-x) = f (x).
From this definition should be a condition necessary for the domain of such a function, namely, the symmetry with respect to point O is the origin, because if a point b contained in the definition of an even function, the corresponding point - b also lies in this area.From the foregoing, therefore, it follows the conclusion: even function is symmetric with respect to the vertical axis (Oy) appearance.
How in practice to determine the parity of the function?
Let the functional relationship is defined by the formula h (x) = 11 ^ x + 11 ^ (- x).Following the algorithm, which follows directly from the definition, we examine first of all its domain.Obviously, it is defined for all values of the argument, that is the first condition is satisfied.
next step we substitute the argument (x) its opposite value (-x).Get
:
h (-x) = 11 ^ (- x) + 11 ^ x.Since
addition satisfies the commutative (commutative) law, then obviously, h (-x) = h (x) and given the functional relationship - even.
verify parity function h (x) = 11 ^ x-11 ^ (- x).Following the same algorithm, we see that h (-x) = 11 ^ (- x) -11 ^ x.Relegate minus, as a result, have
h (-x) = - (x-11 ^ 11 ^ (- x)) = - h (x).Therefore, h (x) - is odd.
way, it should be recalled that there are functions that can not be classified according to these characteristics, they are called either even or odd.
even functions have several interesting properties:
- a result of the addition of these features get even;
- by subtracting these functions get even;
- inverse function even, as the even;
- by multiplying two such functions get even;
- by multiplying the odd and even get the odd functions;
- by dividing odd and even get the odd functions;
- derivative of such a function - an odd;
- if erect odd function in the square, we get even.
parity function can be used to solve the equations.
To solve the equation of g (x) = 0, where the left side of the equation represents the even function, will be enough to find a solution for non-negative values of the variable.These roots must be combined with the additive inverse.One of them is to be checked.
same property function successfully used to solve non-standard problems with a parameter.
For example, if there is any value of the parameter a, for which the equation 2x ^ 6-x ^ 4-ax ^ 2 = 1 will have three roots?
Given that the variable part of the equation in even powers, it is clear that replacing x by - x given equation will not change.It follows that if a number is the root, then it is also the additive inverse.The conclusion is obvious: the roots of non-zero, are included in the set of its solutions "pairs."
clear that the sheer number 0 is not a root of the equation, that is, the number of roots of this equation can only be even and, of course, for any value of the parameter, it can not have three roots.
But the number of roots of the equation 2 ^ x + 2 ^ (- x) = ax ^ 4 + 2x ^ 2 + 2 may 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."We check whether the 0 root.By substituting it into the equation, we obtain 2 = 2.Thus, in addition to "pair" is also the root of 0, which proves their odd number.
