Composite function monotonicity problem, what is the monotonicity of composite function?

The domain of the definition of the outer function is the domain of the value of the inner function.

So you can't equate the range of the preceding outer function with the range of the inner function.

For example, f(g(x)) decrements at x (-1) and increases at (3,+).

t=g(x)

f(t) decrements on t (0,+, where (0,+ is actually the range of g(x).

x (-1) and (3,+ g(x) (0,+ I don't know if you give you a specific function in this question, but if so, that's easy to understand.

In general, composite functions conform to the multiplication rules of Boolean functions.

That is to say: the outer decline and the inner decline, then the negative and negative are positive, which is the increase interval.

The outer increment, the inner decreasing, and the negative positive gain the negative, which is the subtraction range.

Increasing externally, increasingly, increasing the range;

Decreasing outside, increasing inside, negative positive to negative, minus interval.

Isn't it like multiplying -1 and +1?

Now let's verify it

Suppose the inner function is.

g(x)=(x+1)(x-3)

The outer function is: f(x)=1 x

Genius Oh, that all reminds me of "whoami13 smugly narcissist.

Then f(g(x))=1 [(x+1)(x-3)]hoho, answer it yourself

But this seems to be a mistake in the title, and at the very least, it should be said that the foreign function also has a defined domain in (- 1).

The domain of the outer function is only defined in (0,+, so it is impossible to talk about whether it is incremented at (- 1) or not.

The monotonicity of composite functions is "the same increases and the difference decreases". The specific connotation is that if the analytic expression of a composite function is y=f(u(x)), then its outer function is y=f(u) and the inner function is u=u(x).

1) If the monotonicity of the outer function y=f(u) with u as a variable and the inner function with x as variable are the same (same increase or same decrease) in an interval, then y=f(u(x)) is the increasing function on this interval.

2) If the monotonicity of the outer function y=f(u) with u as a variable and the inner function with x as variable are opposite ("inner increase and outer subtraction" or "inner subtraction and outer increase") in an interval, then y=f(u(x)) is the subtraction function on this interval.

The increase or decrease of the above composite function can be simplified into the four cases shown in the following figure with mathematical formulas and symbols

Let the domain of the function y=f(u) be du and the range of values be mu, and the domain of function u=g(x) be dx and the range of mx, if mx du ≠, then for any x in mx du pass u; If there is a uniquely determined value of y, then the variable x and y form a functional relationship through the variable u.

This function is called a composite function and is 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).

Composite functions. The monotonicity of is generally to look at the monotonicity of the two functions contained in the function.

1) If both are incremental, then the function is incremental.

2) One is subtracting and the other is increasing, which is the subtraction function.

3) Both are subtractive, which is the increase function.

Let the function y=f(u) define the domain.

is du, the value range.

is mu, the function u=g(x) defines the domain as dx, the value range is mx, if mx du ≠ , then for any x in mx du through 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).

This is the first book of silver one to ask, the same reason, you make it according to me, I and the silver type, you have the answer, if you have any questions, you can ask directly.

f(x)=2 (2x 2+3) The original function can be split into mega voids: y=2 t (this is an increasing function) t=2 2+3 function t=2x 2+3 opens upward, and the symmetrical axis family burns as: x=0 When x>0, the function t=2x 2+3; monotonically increases, and y=2 t is also monotonically increased, which is increased and subtracted by the co-relatives of the composite function; The original composite function is an increasing function, when x

The monotonicity of composite functions is "the same increases and the difference decreases". The specific connotation is that if the analytic expression of a composite function is y=f(u(x)), then its outer function is y=f(u) and the inner function is u=u(x).

1) If the monotonicity of the outer function y=f(u) with u as a variable and the inner function with x as a variable are the same (same increase or decrease), then y=f(u(x)) is the increasing function on this interval.

2) If the monotonicity of the outer function y=f(u) with u as a variable and the inner function with x as variable are opposite ("inner increase and outer subtraction" or "inner subtraction and outer increase") in an interval, then y=f(u(x)) is the subtraction function on this interval.

The increase or decrease of the above composite function can be simplified into the four cases shown in the following figure with mathematical formulas and symbols

Let the domain of the function y=f(u) be the domain of the god book du and the range of mu and the domain of the function u=g(x) be dx and the range of 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 the zixun y through the variable u.

This function is called a composite function and is 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).

The monotonicity of a composite function is generally based on the monotonicity of the two functions contained in the function.

1) If both are incremental, then the function is incremental.

2) One is subtracting and the other is increasing, which is the subtraction function.

3) Both are subtractive, which is the increase function.

Let the domain of the function y=f(u) be du and the range of values be mu, and the domain of function u=g(x) be dx and the range of 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).

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.

There was a problem with the original answer. The re-answer is as follows:

For the specific even-letter pure muffle number f(x) =x 2 that satisfies the condition of Zhiwang, its monotonicity result should also be the result of this problem.

f(x) =x^2，g(x) =3x^3-7x^2+5, h(x) =f(x-1),h[g(x)] f[g(x)-1] =f(3x^3-7x^2+4) =3x^3-7x^2+4)^2

h[g(x)]}2(3x^3-7x^2+4)(9x^2-14x) =2(3x+2)(x-1)(x-2) ·x(9x-14)

There are a total of 5 stations to make bends, arranged from small to large as -2 3, 0, 1, 14 9, 2

When x x has h[g(x)] draw h[g(x)] sketch as follows:

h[g(x)] monotonically reduced interval is (-2 3), 0, 1), 14 9, 2);

h[g(x)] monotonically increases in the interval (-2 3, 0), 1, 14 9), 2, +

The detailed process is as follows, mainly to investigate the monotonicity of composite functions, and the general idea of this kind of problem is to calculate the intermediate variables u(x) and u'(x) lists the monotonicity interval table of f(u) and u(x). Finally, the conclusion is drawn through the law of "same increase and difference decrease". Wrote some time, I hope it helps, and I like it when I remember the rise of the bucket.

Judgment of monotonicity of composite functions, with"Same increase and different subtraction"。

f(x) is an even function, on (- 0) single subtraction conceals disadvantages, f(x) at (0,+ on single increase, h(x)=f(x-1) on (- 1) single decrease, at (1,+ on single increase, g(x)=3x -7 +5, g (x)=9x -14x, let g (x)<0 get: 0< <14 9, let g (x) 0 get: x<0 or x>14 9, function g(x) on (0,14 9) single decrease, in (- 0), (14 9,+ From the properties of the composite function:

The single increase interval of h(g( ) is pronounced as:

The single reduction interval is: (-slag and trapped, 0), (1, 14 9).

The answer is as follows: cautious return or hunger, more troublesome filial piety:

However, when solving the function interval, the inner function is solved first, so that the composite function is solved layer by layer, and the final solution is obtained to obtain this result.

Here's an example of the detailed steps for the first question:

First, according to the problem circle and the deficit, the expression of the composite shed travel function is solved as follows:

Using the knowledge of derivatives, the main idea is to find the first derivative of the function, and then find the stationing point of the function, so as to judge the monotonicity of the function and find the monotonic increasing and decreasing intervals of the function.

Let f'=0, then:

x1=1, or x2-2x-2=0, i.e. x2,3=1 3

That is, there are three abscissa of the stationary point of the function, and combined with the knowledge points related to inequality and derivatives and the properties of the function, the orange god can find the monotonic interval of the function.

1.The monotonic increase interval is: (1-3,1), (1+3,+2.).The monotonic reduction interval is: (-1-3], [1,1+3].

I really don't understand the knowledge of mathematics, you can find the answer to the answer to find the answer to the state according to the steps, it is best to find a math teacher with a book of Huiguan to give you a lead rock solution, so the accuracy rate will be very high.

1.Finding the Sense Domain of the Composite Function Fixed-Modulus;

2.The composite function is decomposed into several common functions (primary, quadratic source liquid, power, finger split sail, and pair function);

3.judging the monotonicity of each common function;

Solving the monotonicity problem of composite functions Part 3, I really don't understand the knowledge of mathematics, you can find a proposal to find a solution chain loss answer step by step, or find a math teacher who will give you a clear answer step by step.