How to understand that a function is even or odd. Function parity. The largest and smallest value of a function on an interval
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 whether a function is even or odd
1)
;
2)
;
3)
.
Solution.
1) The function is defined when 
. We'll find 
.
Those. 
. This means that this function is even.
2) The function is defined when 
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 function of general form.
3. Study of the function for monotonicity.
Function 
is called increasing (decreasing) on a certain interval if in this interval each larger value of the argument corresponds to a larger (smaller) value of the function.
Functions increasing (decreasing) over a certain interval are called monotonic.
If the function 
differentiable on the interval 
and has a positive (negative) derivative 
, then the function 
increases (decreases) over this interval.
Example 6.3. Find intervals of monotonicity of functions
1)
;
3)
.
Solution.
1) This function is defined on the entire number line. Let's find the derivative.
The derivative is equal to zero if 
And 
. The domain of definition is the number axis, divided by dots 
,
at 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 increases over this interval.
2) This function is defined if 
or
.
We determine the sign of the quadratic trinomial in each interval.
Thus, the domain of definition 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 over the interval 
.
4. Study of the function at the extremum.
Dot 
called the maximum (minimum) point of the function 
, if there is such a neighborhood of the point 
that's for everyone 
from this neighborhood the inequality holds 
.
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 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 a function 
equal to zero 
, and the second derivative exists and is different from zero. If 
, That 
– maximum point, if 
, That 
– minimum point of the function.
Example 6.4 . Explore the maximum and minimum functions:
1)
;
2)
;
3)
;
4)
.
Solution.
1) The function is defined and continuous on the interval 
.
Let's find the derivative 
and solve the equation 
, i.e. 
.From here 
– 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 
– minimum points.
When passing through a point 
the derivative changes sign from “+” to “–”, so 
– maximum point.
,
.
2) The function is defined and continuous in the interval 
. Let's find the derivative 
.
Having solved the equation 
, we'll find 
And 
– critical points. If the denominator 
, i.e. 
, then the derivative does not exist. So, 
– third critical point. Let us determine the sign of the derivative in intervals.
Therefore, the function has a minimum at the point 
, maximum in points 
And 
.
3) A function is defined and continuous if 
, i.e. at 
.
Let's find the derivative 
.
Let's find critical points: 
Neighborhoods of points 
do not belong to the domain of definition, therefore they are not extrema. So, let's examine the critical points 
And 
.
4) The function is defined and continuous on the interval 
. Let's use rule 2. Find the derivative 
.
Let's find critical points:
Let's find the second derivative 
and determine its sign at the points 
At points 
function has a minimum.
At points 
the function has a maximum.
Even function.
Even is a function whose sign does not change when the sign changes x.
x equality holds f(–x) = f(x). Sign x does not affect the sign y.
The graph of an even function is symmetrical about the coordinate axis (Fig. 1).
Examples of an even function:
y=cos x
y = x 2
y = –x 2
y = x 4
y = x 6
y = x 2 + x
Explanation:
Let's take the function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect the 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 changes x.
In other words, for any value x equality holds f(–x) = –f(x).
The graph of an odd function is symmetrical with respect to the origin (Fig. 2).
Examples of odd function:
y= sin x
y = x 3
y = –x 3
Explanation:
Let's take the function y = – x 3 .
All meanings at it will have a minus sign. That is a sign x influences 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 functions are even or odd. There are functions that do not obey 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 that describe these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.
Which were familiar to you to one degree or another. 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 = f(x), x є X, is called even if for any value x from the set X the equality f (-x) = f (x) holds.
Definition 2.
The function y = f(x), x є X, is called odd if for any value x from the set X the equality f (-x) = -f (x) holds.
Prove that y = x 4 is an even function.
Solution. We have: f(x) = x 4, f(-x) = (-x) 4. But(-x) 4 = x 4. This means that for any x the equality f(-x) = f(x) holds, i.e. the function is even.
Similarly, it can be proven that the functions y - x 2, y = x 6, y - x 8 are even.
Prove that y = x 3 ~ an odd function.
Solution. We have: f(x) = x 3, f(-x) = (-x) 3. But (-x) 3 = -x 3. This means that for any x the equality f (-x) = -f (x) holds, i.e. the function is odd.
Similarly, it can be proven that the functions y = x, y = x 5, y = x 7 are odd.
You and I have already been convinced more than once that new terms in mathematics most often have an “earthly” origin, i.e. they can be explained somehow. This is the case with both even and odd functions. See: y - x 3, y = x 5, y = x 7 are odd functions, while y = x 2, y = x 4, y = x 6 are even functions. And in general, for any function of the form y = 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 = 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 = 2x + 3. Indeed, f(1) = 5, and f (-1) = 1. As you can see, here, therefore, neither the identity f(-x) = f ( x), nor the identity f(-x) = -f(x).
So, a function can be even, odd, or neither.
The study of whether a given function is even or odd is usually called the study of parity.
Definitions 1 and 2 refer to the values of the function at points x and -x. This assumes that the function is defined at both point x and point -x. This means that point -x belongs to the domain of definition of the function simultaneously with point x. If a numerical set X, together with each of its elements x, also 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 asymmetric.
– Do even functions have a domain of definition that is 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) – even or odd, then its domain of definition is D( f) is a symmetric set. Is the converse statement true: if the domain of definition of a function is a symmetric set, then is it even or odd?
– This means that the presence of a symmetric set of the domain of definition is a necessary condition, but not sufficient.
– So how do you examine a function for parity? Let's try to create an algorithm.
Slide
Algorithm for studying a function for parity
1. Determine whether the domain of definition 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:
Examine function a) for parity at= x 5 +; b) at= ; V) at= .
Solution.
a) h(x) = x 5 +,
1) D(h) = (–∞; 0) U (0; +∞), symmetric set.
2) h (– x) = (–x) 5 + – x5 –= – (x 5 +),
3) h(– x) = – h (x) => function h(x)= x 5 + odd.
b) y =,
at = f(X), D(f) = (–∞; –9)? (–9; +∞), an asymmetric set, which means the function is neither even nor odd.
V) 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 = x (5 – x 2).
a) y = x 2 (2x – x 3), b) y =
Graph the Function at = f(X), If at = f(X) is an even function.
Graph 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 Unified State Examination option).
1. The odd function y = f(x) is defined on the entire number 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