Why does the composite function increase to increase, increase or decrease to decrease, and decrease

Let a coincidence function be y=f(t), where t=g(x)1, and we are talking about monotonicity buying is talking about the change of the function value y with x.

Both are increments :

The larger the x, the greater the t (because t=g(x) is an increasing function); The larger the t, the greater the y (because y=f(t) is the increasing function), so there is a larger x and the larger y y, so the composite function y is the increasing function.

y=f(t) increases, t=g(x) decreases:

The larger the x, the smaller t (because t=g(x) is a subtractive function); The smaller the t, the smaller the y (because y=f(t) is an increasing function), so there is a greater x and a smaller y, so the composite function is a subtraction function.

In the same way, subtraction becomes increase.

2. Why is the monotonicity different in different intervals?

We know that the monotonicity of the composite function changes with the monotonicity of the integral function, but the monotonicity of the function may be different in different intervals, so the monotonicity of the composite function should also be considered in intervals.

The answer is to be precise, not more, I hope you can understand the problem in essence, otherwise you will still not be able to do the problem yourself no matter how many examples you give. I believe you must have done a lot of questions in this area before asking this question, and the teacher must have talked a lot.

What is an increment function? That is, within a specified interval, when the independent variable increases, the dependent variable also increases.

So, whether it's a composite function or a single function, you just need to see that when x increases, the outermost function y also increases, that's right. To understand that the increase of the composite function is actually to check it in two steps to see if y also increases when x increases. For example:

f(g(x)), assuming that the inner function g(x) increases and the outer function f(x) decreases, then: when x increases, g(x) increases, which is equivalent to taking the first step, but when g(x) increases, f(x)=f(g(x)) decreases, then, from the overall point of view, f(x) decreases when x increases. So, the composite function f(x) is a subtraction function.

Other increases and decreases are similar to the understanding.

Hope you understand.

This is what is called subtraction and increase, if it is a composite function (e.g. composed of two functions), one of which is an increasing function.

The other is also an additive function, so this composite function is an additive function. This is called multiplication and reinforcement.

One is an increase and one is a decrease.

A composite function is a subtractive function.

In fact, it is positive with negative and negative.

Just right, right. Negative is also positive. This is what increases and decreases become subtractions.

One is minus, and the other is also minus.

The composite function is the increase.

Because the monotonicity of composite functions has the following laws:

1 If the outer function is incremental, then the monotonicity of the composite function is the same as that of the inner function.

2 If the outer function is a subtraction function, the monotonicity of the composite function is the opposite of the monotonicity of the inner function.

I wonder if you're satisfied?

Just understand him as.

Just right, right. Positive and negative gain minus.

Negative negative is positive. It's the same thing.

It's too complicated, so I'll teach you how to make it simple.

Break down the function of y about x into separate functions to judge monotonicity, and then according to what you asked, increase to increase, increase to decrease, subtraction to subtraction, you should read the first year of high school now, read math without rote memorization, just understand).

Increment function + increment function = increment function.

Subtractive function + subtractive function = subtractive function.

Increasing function - subtracting function = increasing function.

Subtraction function - increase function = subtraction function.

Increment function - Increment function = uncertain.

Subtractive function - subtractive function = uncertain.

odd function + even function = odd function;

odd function + odd function = odd function;

even function + even function = even function;

Increment function +-increase = increase.

That's right.

In addition, the compound function is the same as the increase and the difference is subtracted, that is, if the monotonicity of the inner and outer functions is the same, then f(x) is the increasing function, otherwise the amount is the decreasing function.

For example, y1=(1 2) x is a subtraction function, y2= x is an increasing function, then the composite function y=(1 2) (x) is a subtraction function, which is "heterosexual subtraction".

For example, y1=(1 2) x is a subtraction function, and y2=1 x is also a subtraction function, so the composite function y=(1 2) (1 x) is an increasing function, which is "the same sex increases".

"Multiply" the rational functions of the external functions (i.e., multiply between +1 and -1), the composite function may sometimes have more than two functions, so the function is subtracted, and if it is a positive cardinal principle, the function is increased; If negative: the same increase and different decreases.

We can think of the increasing function as +1 and the subtraction function as -1

The same increase is an increase, the same decrease is an increase, and an increase and a decrease are a decrease.

The y=f[g(x)] function can be regarded as a composite of two functions, y=f(u) and u=g(x), and is generally called a composite function. where y=f(u) is the outer function and u=g(x) is the inner function.

If the increase or decrease of the inner and outer functions are the same, the original composite function is the increase function. On the contrary, it is a subtraction function, that is, a composite function, and the monotonicity follows the principle of same increase and difference decrease.

In addition, it is important to note that it must be judged within the same range. Don't forget to define the domain, the most important thing about the function is to define the domain.

If it is a composite function (Hikashi is composed of two functions), one of which is an increasing function and the other is also an increasing function, then this composite function is an increasing function. This is called multiplication and reinforcement.

One is an increase and one is a subtraction, and the composite function is the subtraction function. This is what increases and decreases become subtractions.

One is to subtract the other is also to reduce the fibrillation and quickly compound the number of this is to increase. This is what is reduced and what is increased.

In fact, it's the same as negative and negative are positive, positive and positive are positive, and negative is also positive.

Increasing and decreasing is when the inner and outer functions of a composite function have the same monotonicity, and the composite function increases monotonically. Conversely, when the inner function of a composite function is opposite to the outer function monotonicity, the composite function decreases monotonicity.

Determinants

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

Basic steps

Find the defined domain of the composite function;

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

judging the monotonicity of each common function;

Convert the value range of the intermediate variable into the value range of the independent variable;

Find the monotonicity of the composite function.

In general, let the domain of a continuous function f(x) be d, then.

If the values x1, x2 d and x1>x2 of any two independent variables belonging to an interval in the defined domain d have f(x1)>f(x2), i.e., they are monotonionic and monotonically increasing on d, then f(x) is said to be an increasing function in this interval.

Conversely, if the values x1, x2 d, and x1>x2 of any two independent variables that belong to an interval in the defined domain d have f(x1), then the increasing and decreasing functions are collectively referred to as monotonic functions.

Compound function increases and decreases when the inner function of a composite function is monotonionic with the outer function, the composite function increases monotonically, and conversely, when the inner function of a composite function is opposite to the external function monotonicity, the composite function monotonically decreases.

The monotonicity of a function can also be called the addition or decrease of a function. 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 monotonionic in the interval.

The compound function is understood as follows:

It refers to whether the increasing and decreasing properties of two functions are the same when they are compounded. If the increasing and decreasing properties of a composite function are the same as those of one of the functions, but the opposite of the other, it is called the same increase and decrease of the composite function.

1.What is a compound letter in the whispering number:

A composite function is a new function that combines two or more functions through continuous operations. For example, if the functions f(x) and g(x) are both functions over the field of real numbers, then the composite function can be expressed as (f g)(x)=f(g(x)).

2.Understanding of the nature of the same increase:

If both functions have deltas on the same value of the argument variable, i.e. for any x1, x2 (x13.Understanding of the nature of heterosubtraction:

If two functions have an increment and a decrease in the other function on the same independent variable value, i.e., for any x1, x2 (x1g(x2), then these two functions have heterosubtractive properties.

4.Understanding of the same increase and difference of compound functions:

When the functions f(x) and g(x) are combined, if the increasing and decreasing properties of the composite function (f g)(x) are the same as those of one of the functions (i.e., f(x) or g(x) are the same or different subtractions), but the increasing and decreasing properties of the other function are opposite (i.e., the increasing and decreasing properties of f(x) and g(x) are inconsistent), then the composite function is called the same increase and decrease.

Expand your knowledge:

The increasing and decreasing properties of a composite function can be judged by finding a derivative. If both functions are derivative and the derivative sign is the same, then the composite function increases together; If both functions are derivative and the derivative sign is opposite, then there is no compound function subtracted. Compound functions have important applications in mathematical analysis, calculus and other fields, and are used to study the increasing and decreasing properties of functions and optimization problems.

The properties of the compound function can also be observed by means of the function graph. On the function graph, if the rising interval of one function corresponds to the falling interval of another function, it indicates that the composite function increases and decreases.

Summary:

When two functions are compounded, the increasing and decreasing properties of the composite function are the same as those of one of the functions, but the opposite of the increasing and decreasing properties of the other function. This property can be judged by derivation, observation of function images, etc., and has important mathematical application value.

Same increase and different subtraction.

That is, if both functions increase or decrease in the same range, then the composite function is the increase interval; If two functions are in the same range, one plus and one minus, that is the subtraction interval.

No, it is necessary to pay attention to the fact that it must be within the same interval to judge and don't forget to define the domain, the most important thing for the function is to define the domain.

Order x1 x2, by.

g(x) is known as the increasing function, and g(x1) g(x2), so we can make y1=g(x1) and y2=g(x2), then there is y1 y2, and the increasing function f(x) is brought in

f(y1) is obtained

f(y2), i.e. f[g(x1)].

f[g(x2)]

The above is the first article.

f(x)g(x)

f[g(x)]

of the proof of the increase in the increase.

The same can be said for others. Of course, if you understand it, the step "make y1=g(x1), y2=g(x2)" can be omitted.