Demonstrate the monotonicity and parity of abstract functions

I won't teach you parity, the above people have already talked about it completely.

Now I'm going to talk about monotonicity:

Let the definition domain of the x1,x2 function, and x1>x2

Then calculate f(x1)-f(x2) and defactor this formula into its simplest form.

Then calculate the symbol of the equation and compare the magnitudes of f(x1) and f(x2).

If f(x1) > f(x2), it is monotonically increased.

If f(x1)x2

f（x1)-f(x2)

x1^2-2-(x2^2-2)

x1+x2)(x1-x2)

When x [0,+.

x1+x2>0

x1-x2>0

f（x1)>f（x2)

f(x) increases monotonically on [0,+.

When x [0,+.

x1+x2<0

x1-x2>0

f(x1)f(x) is monotonically subtracted on (- 0).

If f(x)=f(-x) is an even function, f(x)=-f(-x) is an odd function.

For example, if f(x) is an odd function, then g(x)=sinf(x) is parity, and g(-x)=sinf(-x)=sin-f(x)=-sinf(x)=-g(x) is an odd function.

Parity. 1 Definitions.

In general, for the function f(x).

1) If there is f(-x)=-f(x) for any x in the function definition field, then the function f(x) is called an odd function.

2) If there is f(-x)=f(x) for any x in the function definition field, then the function f(x) is called an even function.

3) If f(-x)=-f(x) and f(-x)=f(x) are true at the same time as f(-x)=f(x) for any x in the function definition domain, then the function f(x) is both odd and even, and is called both odd and even.

4) If f(-x)=-f(x) and f(-x)=f(x) cannot be true for any x in the function definition domain, then the function f(x) is neither odd nor even, and is called a non-odd and non-even function.

Note: Odd and evenness are integral properties of a function, for the entire defined domain.

The domain of the odd and even functions must be symmetrical with respect to the origin.

If the domain of a function is not symmetric with respect to the origin, then the function must not be an odd (or even) function.

Analysis: To judge the parity of a function, first of all, we should check whether the definition domain is symmetrical with respect to the original resistance macro points, and then simplify and sort it out in strict accordance with the definition of Qichang and evenness, and then compare it with f(x) to draw conclusions).

The basis for judging or proving whether a function is parity is by definition.

2 Parity function image.

Characteristic: Theorem The image of an odd function is centrically symmetrical with respect to the origin.

Charts, images of even functions with respect to the y-axis or axisymmetric graphs.

f(x) is the odd function "f(x)'s image symmetry with respect to the origin.

Point (x,y) (x,-y).

If the odd function increases monotonically over an interval, it also increases monotonically on its symmetrical interval.

The even function increases monotonically over a certain percolation interval and decreases monotonically on its symmetry interval.

Monotonicity: In general, let the function f(x) be defined in the domain i:

If for any two independent variables that belong to an interval within i.

The values of x1 and x2 are f(x1) < f(x2).Then let's say that f(x) is an increasing function in this interval.

If for the values x1 and x2 that belong to any two independent variables on an interval within i, when x1f(x2)Then f(x) is a subtractive function in this interval.

If the function y=f(x) is an increasing or decreasing function in a certain interval. Then let's say that y=f(x) has a (strict) monotonicity in this interval, and this interval is called the monotonic interval of y= f(x), and the image of the increasing function is rising in the monotonic interval, and the image of the subtracting function is decreasing.

Note: (1) The monotonicity of a function is also called the increase or decrease of a function;

2) the monotonicity of a function is a local concept for a certain interval;

3) Method steps for determining the monotonicity of a function over an interval:

a.Let x1 and x2 give a given interval, and x1

Parity is to see that the image of the function is symmetric (even) with respect to the y-axis, i.e. f(x)=f(-x); Or about origin symmetry (odd function), i.e. -f(x)=f(-x).

Monotonicity refers to whether the function image increases or decreases with the increase of x in a certain interval.

I don't know if the explanation is clear enough, you can ask.

Function parity, monotonicity and its discriminant methods.

General function monotonicity discrimination:

1.Definition method: If you set x10 in the definition field, then y increases monotonically; If y'<0 then y is monotonically decreasing.

Parity Discrimination:

1.Definition: Determine parity by calculating f(-x) to determine whether it is equal to f(x) or -f(x).

2.Exploit the properties of the operation: odd = odd odd = even even even = even odd odd = odd even = even.

3.Utilize the derivative:

The derivative of a derivable odd function is an even function.

The derivative of a derivable even function is an odd function.

Monotonicity discrimination of composite functions: the same increase and different decreases. This means that in f(x)=f(g(x)), if f, g have the same monotonicity, then f is an increasing function, and if f, g have different monotonicity, then f is a subtraction function.

Parity of the conforming function: f, g have an even function, f is an even function, only f and g are both odd functions, f is an odd function.

Monotonicity refers to whether a function increases or decreases in a certain interval, i.e. the larger x and the smaller y is.

Whereas, parity refers to symmetry with respect to the y-axis or the origin, where the odd function f(-x) = -f(x).

And the even function f(x) = f(-x).

Looking for courseware on the Internet, the teacher explained it very clearly, including example questions.

Monotonicity: Let x1>x2 (x1, x2 belong to the defined domain and be continuous), compare the size of f(x1) and f(x2), there are two kinds of difference and quotient, if f(x1)> f(x2) is an increasing function, f(x1) parity: if f(x)=f(-x) is an odd function, f(x)=-f(-x) is an even function.

The parity of f(x) cannot be determined and needs to be discussed.

This is a non-odd and non-even function, you only need to find an x to prove that f(x) + f(-x) is not equal to 0 to mean that it is not an odd function, and find an x, f(x)=f(-x) is not true, it means that it is not an even function.

For example, taking x=1, f(1)=(n-k) (n+k), f(-1)=(1-nk) (1+nk), it is obvious that neither odd nor even functions are satisfied.