Finding the monotonicity of a composite function is in doubt

The law of monotonicity:

1) If the functions y=f(u) and u=g(x) are both increasing or decreasing, then the composite function y=f[g(x)] is an increasing function!

2) If one of the functions y=f(u) and u=g(x) is an increasing function and the other is a decreasing function, then the composite function y=f[g(x)] is a subtraction function!

Note: The increase or decrease interval must be within the defined domain!

Example: Determine the monotonicity of y=log3(-3x-2) and find the monotonic interval?

Solution: (1) First, let the intermediate variable: let u=-3x-2, then y=log3(u).

The function defines the domain -3x-2>0 so x<-2 3

u=-3x-2 is a subtraction function on (- 2 3), so if x increases monotonically on (- 2 3), then u monotonically decreases, y=log3(u)(u>0) is an increasing function because the base is greater than 1, and on the interval (- 2 3) u monotonically decreases, then y monotonically decreases.

In summary, it can be seen that in the interval (- 2 3), x monotonically increases, u monotonically decreases, and y monotonically decreases.

Therefore, if x increases monotonically, then y decreases monotonically, so y=f(x)=log3(-3x-2) is a subtraction function on (- 2 3).

Then the subtraction interval of y=log3(-3x-2) is (- 2 3).

General. When x < 0, the deformation determines the positive or negative of the function.

x=0, check whether it is = 0

x>0, the same goes for it.

In special cases, the topic needs you to prove, it needs to be discussed in a classified manner, and sometimes the law is refuted.

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].

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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.

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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

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 monotonicity of the composite function 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". Steps to determine the monotonicity of a composite function.

1. Find the definition domain of the composite function;

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

3. Judge the monotonicity of each common function;

4. Convert the value range of the intermediate variable into the value range of the self-variable reed.

5. Find the monotonicity of the compound function mask.

Explanation of the monotonicity judgment of composite functions.

1. The monotonicity of the function must be discussed within the defined domain, that is, the monotonic interval of the function is a subset of its defined domain, so to discuss the monotonicity of the function, the definition domain of the function must be determined first.

2. The monotonicity of the function is for a certain interval, for a single point, because its function value is the only definite constant, so there is no increase or decrease change, so there is no monotonicity problem; In addition, the main study of continuous functions or piecewise continuous functions in secondary school is that for a continuous function in a closed interval, as long as it is monotonic in the open interval, it is also monotonic in the closed interval, so when considering its monotonic interval, including excluding the endpoints; Also note that for functions that are discontinuous at some points, the monotonic interval does not include discontinuous points.

The monotonicity of the composite function 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".

Steps to determine the monotonicity of a composite functionFind 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;

The value range of the intermediate variable is transformed into the value range of the independent variable of regret.

Find the monotonicity of the composite function.

Explanation of the monotonicity judgment of composite functions. 1. The monotonicity of the function must be discussed within the defined domain, that is, the monotonic interval of the function is a subset of its defined domain, so to discuss the monotonicity of the function, the definition domain of the function must be determined first.

2. The monotonicity of the function is for a certain interval, for a single point, because its function value is the only definite constant, so there is no increase or decrease change, so there is no monotonicity problem; In addition, the main study of continuous functions or piecewise continuous functions in secondary school is that for a continuous function in a closed interval, as long as it is monotonic in the open interval, it is also monotonic in the closed interval, so when considering its monotonic interval, including excluding the endpoints; Also note that for functions that are discontinuous at certain points, monotonic intervals do not include discontinuous points.

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).

1) If both are incremental, then the function is an increment;

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

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

Composite function: let the definition domain of the function y=f(u) be du and the value range be mu, and the definition of the function u=g(x) is dx and the value range is 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 formed by the variable u, which is called a 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= Considering the clever range of the values of x in each part, take their intersection.