Help judge the parity and monotonicity of the function!!

1.Odd + Odd 2Even + Even 3

Odd + Even Any Function [Because any function can be represented as the sum of an odd function and an even function.] Relatively important nature] 4.Odd-odd 5

even-even = even 6Odd-Even = Indefinite 7Even-odd = indefinite 8

Odd*Odd = Even 9Even * Even = Even 10Odd * Even Odd 11

Odd Odd 12Even-even 13Odd Even 14

Odd Odd Odd Odd Even-17- Odd (negative odd function) even 18- Even (negative even function) odd.

Monotness: 1Increase + Increase = Increase 2

minus + minus = minus 3 increase + minus = indefinite 4Increase-increase = indefinite 5minus - minus = indefinite 6

Increase-decrease = increase 7Decrease-increase = increase 8Increase*Increase= Indefinite 9

minus * minus = indefinite 10Increase*Decrease = Indefinite 11Increase = Indefinite 12

minus = indefinite 13Increase or decrease = indefinite 14Subtraction = Indefinite increase = Decrease = Increase 17

Increase (negative increase function) = minus 18- Subtraction (negative subtraction function) = increase

There may be errors. Because there are some questions that are not asked often. May not be considered in detail.

You take -3, -2, -1, 1, 2, 3 and you know, I'm dazzled.

The easiest way to use.

Derivatives to distinguish.

Step: Parity:

1.Let's see if the definition domain is symmetrical with respect to the origin.

2.If it is not closed back to the origin symmetry, then the function does not answer parity3If the domain is defined symmetrically with respect to the origin.

4.then f(-x) = f(x), where f(x) is an even function and f(x) is an odd function.

Monotness: 1First, take two values on the interval, generally x1 and x2 and set x1 x2 (or x1 x2).

2.Substituting x1 and x2 into the f(x) analytic formula to make the difference, that is, f(x1)-f(x2).

3.Simplified, multiplied or divided.

4.If f(x1)-f(x2) 0 is satisfied, it is an increment function.

1. If the base is the same and the exponents are different, use the monotonicity of the exponential function to do it;

2. If the exponents are the same but the bases are different, draw an image of the two functions, such as judgment and.

First draw the image of f(x)=, g(x)=, and observe the height of the function image of x= to judge the size of the function value;

In fact, this can indeed be done with a power function (I guess I learned it in a few weeks) to judge monotonicity (this beam collapse may sometimes involve derivative problems, which is an elective content in the third year of high school).

Third, the index is different, the base is also different, find the middle amount, usually 1However, it is not excluded that others, such as interpretation, and 1 judgment, both result in a smaller than 1, so choose another intermediate quantity to do.

The previous method can be used, and now the general method is supplemented, and the derivative needs to be used).

For example, there are two numbers: a e and e a, a> e, e is approximately equal, * is the multiplication sign, a ea is the base e is the referent.

Let a e = a, e a = b, f(x) = e x-x e

Hence f'(x)=e^x-e*x^e-1

Because xmin=e, x=0, f'(x) > 0, so f(x) increments on (0,e).

Because x>e, f'(x) <0, so the oak circle f(x) decreases on (e, positive infinity).

Because when x=e, f(x)=0

Because a>e, substituting a into f(x)b

This question is encountered by the respondent when doing the question, although it is a bit unusual, but it is enough as a reference. In addition, the comparison of the logarithm and then the bottom of the base as mentioned on the Internet is not feasible in this question. }

The method of judging the senileness and parity of the rock cluster is as follows.

The methods for judging the monotonicity of functions include the definition method, the property method, the compound function addition and subtraction method, and the derivative method.

Parity is generally judged by drawing, and other methods are to use coarse liter definitions and function operations.

Monotonicity means that when the independent variable of the function f(x) increases (or decreases) within its defined interval, and the value of the function f(x) also increases (or decreases), the function is said to be monotonicity in the interval.

Parity is one of the fundamental properties of functions.

In general, if there is f(-x)=f(x) for any x in the definition domain of the function f(x), then the function f(x) is called an even function.

In general, if there is f(-x)=-f(x) for any x in the domain where the function f(x) is defined, then the function f(x) is called an odd function.

Function parity, monotonicity and its discriminant methods.

General function monotonicity discrimination:

1.Definition method: If the x1 answer is greater than 0 in the definite return domain, it will increase monotonically; If it is less than 0, it decreases monotonically.

2.Derivative method: Derivative of the derivative function y=f(x) if y'>0, 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 increases, and the difference 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.

Monotness: 1Judging by the monotonicity of the basic function.

2.Derivation. 3.

According to the monotonicity of the composite function, i.e., the same increases and decreases. 4.Judgment within with f(x1).

f(x2). 5.If it is a pumping function, you should set x2>x1 and write f(x1) according to the known conditions

f(x2).

Parity: Look at f(x).f(-x).

If f(x)=f(-x), it is an even function, and if -f(x)=f(-x), it is an odd function. If it is a logarithmic function, you should use the logarithmic function addition, use f(x)+f(-x)=0 or f(x)-f(-x)=0, and determine the parity after shifting the term.

If a piece of math can't be solved, the easiest way to score is to solve it with definitions, so take a good look at the definitions of each knowledge point in the book.