Parity of composite functions, parity of composite functions How to judge

You can think about a few functions

It can also be rigorously proven by definition.

For example: odd g(x)=-g(-x) even f(x)=f(-x) composite function: f(g(x))=f(-g(-x))=f(g(-x))) even function.

If you guessed correctly.

The answer is: odd and even.

As long as there is an even function in the composite function, the composite function is an even function, such as an odd and an even function;

If there are only odd functions, the composite function is an odd function, regardless of odd or even number, such as two odd functions are still odd.

1. f(x)*g(x)*h(x).

The number of odd functions is even, and the composite function is even.

The number of odd functions is odd, and the composite function is odd.

2. f(g(h(x))) is a multi-layered composite function.

There are even numbers in a function, and a composite function is an even function.

There are no even numbers in functions, the number of odd functions is even, and composite functions are even functions.

There are no even numbers in functions, the number of odd functions is odd, and composite functions are odd functions.

This can be demonstrated by the example of the special law.

It can also be justified by definitions. If f(x) is odd and g(x) is odd, then f(g(x)) is odd.

It's best if you can prove it yourself with definitions or properties, and knowing the results will not help you to do the proof questions.

I forgot, but the compound depends on what kind of compound method it is, right?

The questions in high school are really hard.

Let's start with compounding.

Definition domain of the function:

If the definition domain is not symmetric with respect to the origin, then the composite function is a non-odd and non-even answer.

Number; If the domain is defined with respect to origin symmetry, then look at the inner and outer functions:

When the inner function is an even function, no matter what kind of function the outer function is, the composite function must be an even function;

When the inner function is an odd function and the outer function is also an odd function, the composite function is an odd function;

When the inner function is an odd function and the outer function is an even function, the composite function is an even function.

Outside odd and inside odd is odd, outside odd and inner odd is even, outside even and inside odd is even, outside even and inside even is even.

f=f(g(x)), if g(x) is an even function, when arbitrarily taking two points x1,-x1 with respect to x symmetry, there is g(x1)=g(-x1), so f(g(x1))=f(g(-x1)). f is an even function, so the inner even is even. f=f(g(x)), if g(x) is an odd function, when arbitrarily taking two points x1,-x1 with respect to x symmetry, there is -g(x1)=g(-x1), so when f is even, f(-g(x1))=f(g(-x1)) then the whole is even.

When f is odd, -f(-gx1))=-f(g(-x1)) is odd overall.

Let the domain of the function y=f(x) be du and the value range be mu, and the domain of the function u=g(x) be dx and the range be mx, if mx du ≠, then for any x in mx du pass u; If there is a uniquely determined value of y, then there is a functional relationship between the variable x and y through the variable u, which is called the composite function, denoted as: y=f[g(x)], where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function).

If the domain of the function y=f(u) is b and the domain of u=g(x) is a, then the domain of the composite function y=f[g(x)] is .

d= Take the range of x values of each part and take their intersection.

Finding the definition domain of a function should mainly consider the following points:

the range of r when it is an integer or an odd root form;

When it is an even radical, the number of squares to be opened is not less than 0 (i.e., 0);

When it is a fraction, the denominator is not 0; When the denominator is an even radical, the number of squares to be opened is greater than 0;

When exponential, the base is not 0 for the exponential power of zero or negative integer exponential power (e.g., medium).

When it is formed by combining some basic functions through four operations, its definition domain should be the set of values of independent variables that make each part meaningful, that is, find the intersection of the set of definition domains of each part.

The definition domain of a piecewise function is the union of the set of values of the independent variables on each segment.

Functions built by practical problems should consider not only the requirements of the arguments for the analytic expression, but also the requirements of the arguments for the practical meaning.

For functions with parameter letters, the values of the letters should be classified and discussed when finding the definition domain, and it should be noted that the definition domain of the function is a non-empty set.

The true number of the logarithmic function must be greater than zero, and the base number must be greater than zero and not equal to 1.

The cutting function in trigonometric functions should be careful to pay attention to the limitation of diagonal variables.

Let the minimum positive period of y=f(u) be t1, and the minimum positive period of = (x) be t2, then the minimum positive period of y=f( ) is t1*t2, and any period can be expressed as k*t1*t2 (k belongs to r+).

It is determined by the monotonicity of y=f(u), = (x). Namely"increase + increase = increase; minus + minus = increase; increase + decrease = decrease; minus + increase = minus", which can be simplified to:"Same increase and different subtraction"。

In fact, as long as you grasp the definition of a curious even function, it is very easy to push it yourself. Here are some examples:

Write f(x)=f[g(x)]—composite function, then f(-x)=f[g(-x)]

If g(x) is an odd function, i.e., g(-x)=-g(x) ==f(-x)=f[-g(x)], then when f(x) is an odd function, f(-x)=-f[g(x)]=-f(x), and f(x) is an odd function;

When f(x) is an even function, f(-x)=f[g(x)]=f(x), and f(x) is an even function.

If g(x) is an even function, i.e. g(-x)=g(x) =f(-x)=f[g(x)]=f(x), f(x) is an even function.

Therefore, a composite function composed of two functions, when the function of the inner layer is an even function, the even function of the composite function, regardless of the function of the outer layer; When the inner function is odd and the outer function is also odd, the composite function is an odd function, and when the inner function is an odd function and the outer function is an even function, the composite function is an even function.

In other cases, it is not possible to judge the parity of a composite function.

The parity of composite functions is characterized by the following: "the inner even is even, and the inner odd is the same."

outside". f(g(x)), if g(x) is an even function, when arbitrarily taking two points x1 and -x1 symmetrically with respect to x, there is g(x1)=g(-x1), so f(g(x1))=f(g(-x1)). Therefore, the inner couple is even.

In fact, as long as you grasp the definition of a curious even function, it is very easy to push it yourself.

Write f(x)=f[g(x)]—composite function, then f(-x)=f[g(-x)], if g(x) is an odd function, i.e., g(-x)=-g(x) ==> f(-x)=f[-g(x)], then when f(x) is an odd function, f(-x)=-f[g(x)]=-f(x), f(x) is an odd function;

When f(x) is an even function, f(-x)=f[g(x)]=f(x), and f(x) is an even function.

If g(x) is an even function, i.e. g(-x)=g(x) =f(-x)=f[g(x)]=f(x), f(x) is an even function.

Therefore, a composite function composed of two functions, when the function of the inner layer is an even function, the even function of the composite function, regardless of the function of the outer layer; When the inner function is odd and the outer function is also odd, the composite function is an odd function, and when the inner function is an odd function and the outer function is an even function, the composite function is an even function.

In other cases, the parity of the composite function cannot be determined.

No matter how many layers a composite function has, only if all layers are odd, the composite function is an odd function, and as long as one or more layers are even functions, the composite function is even.

If you want to judge the parity of composite functions, this thing is still quite annoying, you can try it.

Odd functions compound odd functions are odd functions;

Odd functions are compound even functions;

Even function composite even function is even function;

Even functions are compound odd functions are even functions;

The product (or quotient) of two odd functions is an even function; The product (or quotient) of two even functions is an even function; The product (or quotient) of an odd and even function is an odd function; The sum and difference of two odd functions (or two even functions) are odd functions (or even functions).

General Principle: Odd Functions f(-x) = -f(x), Even Functions f(-x) = F(x) Odd Functions * Odd Functions = Even Functions, Odd Functions * Even Functions = Odd Functions Even Functions * Even Functions = Even Functions, Odd Functions + - Odd Functions = Odd Functions Even Functions + - Even Functions = Even Functions.

If the inner and outer functions are even, then the composite function is the even function.

There is no parity in the addition and subtraction of odd-even functions.

Not necessarily! Some questions can't be judged.

The key is to grasp the relationship between f(-x) and f(x) and -f(x), and the problem can be made no matter how it comes out.

Compound function parity formula: the outer odd is the odd inside the odd, the outer odd is the even inside the even, the outer even the inner odd is even, the outer even and the inner even is even.

Judging the parity of composite functions:

If g(x) is an odd function, i.e., g(-x)=-g(x) =f(-x)=f, then when f(x) is an odd function, f(-x)=-f=-f(x), f(x) is an odd function;

When f(x) is an even function, f(-x)=f=f(x) and f(x) is an even function.

If g(x) is an even function, i.e. g(-x)=g(x) =f(-x)=f=f(x), f(x) is an even function.

Therefore, a composite function composed of two functions, when the function of the inner layer is an even function, the even function of the composite function, regardless of the function of the outer layer; When the inner function is odd and the outer function is also odd, the composite function is an odd function, and when the inner function is an odd function and the outer function is an even function, the composite function is an even function.

Monotonicity judgment of composite functions:

1. Find the definition domain of the composite function;

2. Decompose the composite function into several common functions (primary, quadratic, power, finger, and pair functions);

3. Judge the monotonicity of each common function;

4. Convert the value range of intermediate variables into the value range of independent variables;

5. Find the monotonicity of the composite function.

Composite function parity.

How to judge:

1. The domain of the function.

There must be symmetry with respect to the origin so that the function can be parity.

2. Definition method: x belongs to the definition domain a of the function y=t(x), and x belongs to the condition of a.

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

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

If f(-x)=-f(x)=f(x)=o, then y=f(x) is an even function and an odd function;

If f(-x)=-f(x)=f(x) is equal to a constant that is not zero, then y=f(x) is an even function.

3. Judge according to the symmetry of the function image: if the function image is symmetrical with respect to the origin, it is an odd function, and if the function image is symmetrical with respect to the y-axis, it is an even function.

4. Judgment of parity of piecewise functions: It is necessary to look at the relationship between f(-x) and f(x) on each segment, or to take the absolute value symbol and simplify the function.

5. Determination of parity of composite functions: function y=f(t) and t=g(x), if f(t) is an odd (even) function, then t=g(x) is an odd (even) function.

6. It is an inverse function of each other.

Relationship judgment: If a function is an odd function, then its inverse function is also a starting function, but even functions cannot have such a relationship.

7. Use special values to judge the parity of functions.

The parity of composite functions is characterized by the following: "the inner is even, and the inner odd is the same as the outside". f(g(x)), if g(x) is an even function, when arbitrarily taking two points x1 and -x1 symmetrically with respect to x, there is g(x1)=g(-x1), so f(g(x1))=f(g(-x1)).

Therefore, the inner couple is even.

f(g(x)), if g(x) is an even function, when arbitrarily taking two points x1 and -x1 symmetrically with respect to x, there is g(x1)=g(-x1), so f(g(x1))=f(g(-x1)). Therefore, the inner couple is even.

f(g(x)), if g(x) is an odd function, when two points x1 and x2 are arbitrarily taken with respect to x symmetry, there is -g(x1)=g(-x1), so when f is even, f(g(x1)) = f(-g(x1)) = f(g(-x1)) then the whole is even. When f is odd, f(g(x1))) = f(-g(x1))) = f(g(-x1)) is odd overall.

f(x)=f[g(x)]—composite function, then f(-x)=f[g(-x)], if g(x) is an odd function, i.e., g(-x)=-g(x)==f(-x)=f[-g(x)], then when f(x) is an odd function, f(-x)=-f[g(x)]=f(x), f(x) is an odd function;

When f(x) is an even function, f(-x)=f[g(x)]=f(x), and f(x) is an even function.

If g(x) is an even function, i.e. g(-x)=g(x) =f(-x)=f[g(x)]=f(x), f(x) is an even function.

Therefore, a composite function composed of two functions, when the function of the inner layer is an even function, the even function of the composite function, regardless of the function of the outer layer; When the inner function is odd and the outer function is also odd, the composite function is an odd function, and when the inner function is an odd function and the outer function is an even function, the composite function is an even function.