The question of sets and functions, how does a function relate to a set

This is the problem of finding the expression of f(x) from the expression of f[g(x)], and the commonly used methods include matching method, commutation method, undetermined coefficient method, identity method, etc. As in this question:

1. Matching method:

f(2x+1)=x²-2x=(1/4)(2x+1)²-3/2)(2x+1)+5/4，∴f(x)=x²/4+3x/2+5/4。

2. Substitution method:

Let y=2x+1, then x=(y-1) 2, f(y)=[(y-1) 2] -y-1)=y 4-3y 2+5 4, so f(x)=x 4+3x 2+5 4.

3. Pending coefficient method: more complicated and omitted.

4. Identity method:

If you want to know, I can tell you too.

f(2x+1)=x2-2x

Let y=2x+1

x=(y-1)/2

f(y)=[(y-1)/2]²-y-1)

y²/4-3y/2+5/4

So. f(x)=x 4+3x 2+5 4 I wish you progress in your studies.

Functions describe the dependence of quantities in nature, reflecting the relationship and laws of one thing changing with the change of another thing The essence of the idea of function is to eliminate the non-mathematical features of the problem, put forward mathematical objects with the observation of connections and changes, abstract their mathematical characteristics, and establish functional relations When solving some numerical problems, we first set some unknowns, and then treat them as known numbers, and list the equations according to the constraints between the quantities of the problem itself, and the unknowns communicate the relationship between the variables. This is the idea of equations Functions and equations are two different concepts, but they are closely related, if a function has an analytic expression, then this expression can be regarded as an equation A binary equation, there is a correspondence between two variables, if this correspondence is a function, then this equation can be regarded as a function, a unary equation, its two ends can be regarded as functions respectively, and the solution of the equation is the abscissa of the intersection of the two function images, therefore, Many problems with equations can be solved by functional methods; On the contrary, many problems related to functions can be solved by means of equations In short, in the review, we should pay attention to comprehending the ideas of functions and equations contained in the knowledge and problem solving process, and use it to guide the solution In the problem solving, we should pay attention to the modification observation and exploration from different angles, and seek a variety of methods, so as to get the best solution You go to search for the set of functions should be able to search out, I am inconvenient to give you this way, and many symbols do not know how to play.

From the set of points in q, 9 points can be obtained by enumeration.

Q = re-analyze the function characteristics of p: let Z = 2 (X a) This function must pass through the point x + a = 1, z = 0, and substitute a to get three points, that is, (3 2, 0), (1 2, 0), and f(x) = z + b, substitute b to get 9 cases: (3 2, -1), (3 2, 0), (3 2, 1), (1, -1), (1, 0), (1, 1), (1 2, -1), (1 2, 0), Judging from the trend of the function graph passing through these points, it is impossible to have a case of (3 2,-1), (3 2,0), (3 2,1), (1,-1); It is also impossible for (1,1) to pass through the points in the set of two qs, so there are only 4 cases that can satisfy the requirements (1,0), (1 2,-1), (1 2,0), (1 2,1), so the answer is a

Take your time and bring it in! Theoretically, it says 144 times, but not because the logarithmic function of the true number must be greater than 0

All three BCD functions are monotonically increasing functions, so they all have no upper bound.

The function y = 1 2) x, which is monotonically decreasing, the larger x is, the closer y is to 0, that is, y > 0, but any value greater than 0 is in the range of y, so 0 is the exact value of the function. But when x decreases along the negative axis, y increases infinitely, so there is no upper boundary.

But a is y = 1 2) x, and the function y = 1 2) x is symmetric on the x-axis, and from the above discussion, it can be seen that the function a is a monotonically increasing function, and the larger the x, the closer y is to 0, that is, y < 0 and any value smaller than 0 is in the range of y, so 0 is the upper definite value of the function a.

By definition, it is known that this function y=f(x) m, where m is a real number.

Of the above functions, only the function in a satisfies this requirement. You can draw an image to feel it intuitively.

In a, y is constant less than 0, that is, m is greater than or equal to 0

It is possible to ignore the less than or equal to the equal in the question).

Solution: m:(x-2) -y-2) <=2

n:x>=y or x+y<=0

The first one is a circle centered on (2,2) and the root number 2 is the radius, and the second image should be easy to draw, and the intersection is what you are looking for.

Solution: f(x)=x 4x 3 f(x) f(y) 0 i.e.: x 4x 3 y 4y 3 0 (x—2) y 2) 2 (draw the figure yourself) At this point, the area represented by the set m is:

A circle (including the circumference) surrounded by the center of the circle (2,2) and a circle of radius 2. f(x) f(y) 0 i.e.: x 4x 3 y 4y 3 0 (x—2) y—2) 0 x 2 y 2 In this case, the area represented by the set n is:

The area to the left and right where the line x-y=0 intersects with the straight line x-y-4=0... As shown in the figure, the area represented by m n is the left and right sector area sandwiched by the intersection of the straight line x-y=0 and the straight line xy-4=0.

It is also easy to know that the two straight lines are perpendicular, that is, the circumferential angles of the arcs of the two sectors are equal. The areas of the two sectors are equal, the whole circle is divided into four congruent pieces, and the sum of the areas of the two sectors is half of the area of the circle.

That is, the area of the set m n is s=(1 2)· 2) =

Solution: When x<0, f(x) increases monotonically, and f(-1)=0 then when -10 is 00, g(0)2 3}

From the above solution, we can get that when x (0, 2), g(x)<-1, n1=;

When x (0, Xun Cong 2), g(x) (0, 1) at this time. When -10, g(0)4 3}

The function of f here is to give a corresponding law. For example, f(x)=3x 2+2x-1, which means that for any number in the set of a, each number in a becomes the corresponding number in b by the action of f (i.e., squared, multiplied by 3, plus twice the number and subtracted by one).

Actually, to be precise, the function is f, and f(x) is not called a function, but a function value.

f represents a correspondence in which a uniquely determined number is obtained given a numerical value in a defined domain.

Sometimes f can be described in mathematical functions, and some can only be described in mathematical language, not necessarily a specific function.

Here, f is a specific calculation method, given x=5, the corresponding definite number according to the relationship is a number, and f is a function.

Hello! The definition of a function is really not easy for beginners to understand, so you need to think carefully and digest it slowly!

The function of f here is to give a law of correspondence, and in layman's terms, f is to stipulate a rule of operation, just like we stipulate the square of numbers, the power of the number, and so on

It is better to say: f(x)=3x 2+2x-1, which means that for any number in the set of a number, according to the operation rules prescribed by f (i.e., squared, multiplied by 3, plus twice the number and then subtracted by one), each number in a becomes the corresponding number in b.

At the specific point, when x = 5 a, the operation according to the operation rules specified by f is 3*(5 2)+2*5-1=84 b

That is, the number 5 in a becomes the number 84 in b through f!

It is known that the complement of a is equal to b, and the complement of a is equal to b: b belongs to the complement of a, and a complement =

Thus: a=

2 and 3 are the roots of the equation x2+ax+b=0.

Bring in, get 4-2a+b=0 9+3a+b=0a=-1, b=-6

Again: b belongs to the complement of a, and a complement =

It can be seen that the range of the solution of the equation kx2+4x+k+3>=0 belongs to k<0 (first the opening should be downward).

And -2 and 3 substitution to kx2+4x+k+3 must be satisfied: kx2+4x+k+3<=0 (you can know the drawing).

There is a system of equations: 4k-8+k+3<=0

9k+12+k+3<=0

k<0 thus obtains: k<=-3 2

By the way, it is also forgotten that the central axis of the parabola kx2+4x+k+3 should also fall in the range i.e.: -2<=-2 k<=3 k<0.

k<=-2/3

Combined: k<=-3 2

Also, this is my high school education, and now that it's been a year, the process may not be written in a standardized way, but the meaning should be correct.

1. x is relative to the sea level, for every 100 meters of altitude, the temperature will drop, then for every 1 meter of height, the temperature will drop, and for every x meters of altitude, the temperature will drop.

So the relationship between temperature and height is: t(x)=

If x is considered to be the height of the world, the analytic formula is not like this.

2. Using the form t(x) to represent the function, it is easier to see who is the independent variable and who is the function. For example, when there are several letters in an analytic formula, this expression can know which is a variable and which is a constant. This expression will be convenient to use later.

Take your time to get used to it.

3. The definition of the range and the value range can be written in the form of a set or an interval. For example, [0,7500] is the interval expression. The domain is defined as: , and this expression is a collection mode, both of which are true.

4. The definition domain is the value range of height x, x is relative to sea level, the minimum height is 0, and the highest will not exceed the height of the mountain 7500 meters, so the first scatter definition domain is: [0, 7500].

f(x+a) is meaningful, 1 x+a 8, -a x 8-a, f(8x+a) is meaningful, 1 8x+a 8, -a 8 x (8-a) 8, both are meaningful, and should satisfy -a Traveler's Rock 8 x (8-a) 8, which is the domain of f(x).