A function is called even (odd) if for any and the equality 
.
The graph of an even function is symmetrical about the axis 
.
The graph of an odd function is symmetrical about the origin.
Example 6.2. Examine for even or odd functions
1)
;
2)
;
3)
.
Decision.
1) The function is defined with 
. Let's find 
.
Those. 
. So this function is even.
2) The function is defined for 
Those. 
. Thus, this function is odd.
3) the function is defined for , i.e. for
,
. Therefore, the function is neither even nor odd. Let's call it a general function.
3. Investigation of a function for monotonicity.
Function 
is called increasing (decreasing) on some interval if in this interval each larger value of the argument corresponds to a larger (smaller) value of the function.
Functions increasing (decreasing) on some interval are called monotonic.
If the function 
differentiable on the interval 
and has a positive (negative) derivative 
, then the function 
increases (decreases) in this interval.
Example 6.3. Find intervals of monotonicity of functions
1)
;
3)
.
Decision.
1) This function is defined on the entire number axis. Let's find the derivative.
The derivative is zero if 
and 
. Domain of definition - numerical axis, divided by points 
,
for intervals. Let us determine the sign of the derivative in each interval.
In the interval 
the derivative is negative, the function decreases on this interval.
In the interval 
the derivative is positive, therefore, the function is increasing on this interval.
2) This function is defined if 
or
.
We determine the sign of the square trinomial in each interval.
Thus, the scope of the function
Let's find the derivative 
,
, if 
, i.e. 
, but 
. Let us determine the sign of the derivative in the intervals 
.
In the interval 
the derivative is negative, therefore, the function decreases on the interval 
. In the interval 
the derivative is positive, the function increases on the interval 
.
4. Investigation of a function for an extremum.
Dot 
is called the maximum (minimum) point of the function 
, if there is such a neighborhood of the point 
that for everyone 
this neighborhood satisfies the inequality 
.
The maximum and minimum points of a function are called extremum points.
If the function 
at the point 
has an extremum, then the derivative of the function at this point is equal to zero or does not exist (a necessary condition for the existence of an extremum).
The points at which the derivative is equal to zero or does not exist are called critical.
5. Sufficient conditions for the existence of an extremum.
Rule 1. If during the transition (from left to right) through the critical point 
derivative 
changes sign from "+" to "-", then at the point 
function 
has a maximum; if from "-" to "+", then the minimum; if 
does not change sign, then there is no extremum.
Rule 2. Let at the point 
first derivative of the function 
zero 
, and the second derivative exists and is nonzero. If a 
, then 
is the maximum point, if 
, then 
is the minimum point of the function.
Example 6.4 . Explore the maximum and minimum functions:
1)
;
2)
;
3)
;
4)
.
Decision.
1) The function is defined and continuous on the interval 
.
Let's find the derivative 
and solve the equation 
, i.e. 
.from here 
are critical points.
Let us determine the sign of the derivative in the intervals , 
.
When passing through points 
and 
the derivative changes sign from “–” to “+”, therefore, according to rule 1 
are the minimum points.
When passing through a point 
derivative changes sign from "+" to "-", so 
is the maximum point.
,
.
2) The function is defined and continuous in the interval 
. Let's find the derivative 
.
By solving the equation 
, find 
and 
are critical points. If the denominator 
, i.e. 
, then the derivative does not exist. So, 
is the third critical point. Let us determine the sign of the derivative in intervals.
Therefore, the function has a minimum at the point 
, maximum at points 
and 
.
3) A function is defined and continuous if 
, i.e. at 
.
Let's find the derivative 
.
Let's find the critical points: 
Neighborhoods of points 
do not belong to the domain of definition, so they are not extremum t. So let's explore the critical points 
and 
.
4) The function is defined and continuous on the interval 
. We use rule 2. Find the derivative 
.
Let's find the critical points:
Let's find the second derivative 
and determine its sign at the points 
At points 
function has a minimum.
At points 
function has a maximum.
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.
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.
Decision. 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.
Decision. 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 ; (∞;∞) are symmetric sets, and , [–5;4] are nonsymmetric.
- Do even functions have a domain of definition - a symmetric set? The odd ones?
- If D( f) is an asymmetric set, then what is the function?
– Thus, if the function at = f(X) is even or odd, then its domain of definition is D( f) is a symmetric set. But is the converse true, if the domain of a function is a symmetric set, then it is even or odd?
- So the presence of a symmetric set of the domain of definition is a necessary condition, but not a sufficient one.
– So how can we investigate the function for parity? Let's try to write an algorithm.
Slide
Algorithm for examining a function for parity
1. Determine whether the domain of the function is symmetrical. If not, then the function is neither even nor odd. If yes, then go to step 2 of the algorithm.
2. Write an expression for f(–X).
3. Compare f(–X).and f(X):
- if f(–X).= f(X), then the function is even;
- if f(–X).= – f(X), then the function is odd;
- if f(–X) ≠ f(X) and f(–X) ≠ –f(X), then the function is neither even nor odd.
Examples:
Investigate the function for parity a) at= x 5 +; b) at= ; in) at= .
Decision.
a) h (x) \u003d x 5 +,
1) D(h) = (–∞; 0) U (0; +∞), symmetric set.
2) h (- x) \u003d (-x) 5 + - x5 - \u003d - (x 5 +),
3) h (- x) \u003d - h (x) \u003d\u003e function h(x)= x 5 + odd.
b) y =,
at = f(X), D(f) = (–∞; –9)? (–9; +∞), asymmetric set, so the function is neither even nor odd.
in) f(X) = , y = f(x),
1) D( f) = (–∞; 3] ≠ ; b) (∞; –2), (–4; 4]?
Option 2
1. Is the given set symmetric: a) [–2;2]; b) (∞; 0], (0; 7) ?
a); b) y \u003d x (5 - x 2).
a) y \u003d x 2 (2x - x 3), b) y \u003d
Plot the Function at = f(X), if at = f(X) is an even function.
Plot the Function at = f(X), if at = f(X) is an odd function.
Mutual check on slide.
6. Homework: №11.11, 11.21,11.22;
Proof of the geometric meaning of the parity property.
*** (Assignment of the USE option).
1. The odd function y \u003d f (x) is defined on the entire real line. For any non-negative value of the variable x, the value of this function coincides with the value of the function g( X) = X(X + 1)(X + 3)(X– 7). Find the value of the function h( X) = at X = 3.
7. Summing up
