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As far as the relationship is concerned, it is generally two-way, and so is the function. Let y = f (x) be a known function. If every y has a unique x∈X, so that f (x) = y, it is the process of finding x from y, that is, x becomes a function of y, and it is recorded as x = f-65438. F-1 is the inverse function of F. Traditionally, X is used to represent the independent variable, so this function is still recorded as y = f- 1 (x). For example, y = sinx and y = arcsinx are reciprocal functions. In the same coordinate system, the graphs of y = f (x) and y = f- 1 (x) are symmetrical about the straight line y = x.

implicit function

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If the function equation F(x, y) = 0, it can be determined that y is a function Y = F(x) y=f(x, that is, F(x, f(x))≡0, then y is said to be an implicit function of X.

Thinking: Is implicit function a function? Because in the process of its reform, it is not satisfied with "one-on-one" and "many-on-one"

Multivariate function

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Setpoint (x 1, x2, …, xn) ∈G? Rn，U？ R 1, if there is a unique u∈U corresponding to each point (x 1, x2, ..., xn)∈G f: g→ u, u = f (x 1, x2, ..., xn.

Basic elementary functions and their images such as power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function are called basic elementary functions.

① Power function: y = x μ (μ ≠ 0, μ is any real number) Definition domain: μ is a positive integer: (-∞, +∞), μ is a negative integer: (-∞, 0)∞(0, +∞); μ = (α is an integer), (-∞, +∞) when α is odd, and (0, +∞) when α is even; μ = p/q, p, q is coprime, and is a composite function of. Sketches are shown in Figures 2 and 3.

② exponential function: y = ax (a > 0, a≠ 1), defined as (-∞, +∞), with a range of (0, +∞), which is strictly monotonic when a > 0 (i.e. x2 > x 1, 0 < a). For any A, the image passes through the point (0, 1). Note that the graphs of y = ax and y = () x are symmetrical about y axis. As shown in figure 4.

③ logarithmic function: y = logax (a > 0), where a is the base, the domain is (0, +∞), and the range is (-∞, +∞). A > 1 strictly monotonically increases, and 0 < A < 1 strictly monotonically decreases. No matter what the value of a is, the graph of logarithmic function passes through the point (1, 0), and both logarithmic function and exponential function are reciprocal functions. As shown in fig. 5.

Logarithms with the base of 10 are called ordinary logarithms and abbreviated as lgx. Logarithm based on e, that is, natural logarithm, is widely used in science and technology, and is recorded as lnx.

④ Trigonometric function: See Table 2.

Sine function and cosine function are shown in figs. 6 and 7.

⑤ Inverse trigonometric function: See Table 3. Hyperbolic sine and cosine are shown in figure 8.

⑥ Hyperbolic function: hyperbolic sine (ex-e-x), hyperbolic cosine? (ex+e-x), hyperbolic tangent (ex-e-x)/(ex+e-x), hyperbolic cotangent (ex+e-x)/(ex-e-x).

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In the field of mathematics, a function is a relationship, which makes each element in one set correspond to the only element in another (possibly the same) set (this is only the case of unary function f (x) = y, please give a general definition according to the original English text, thank you). The concept of function is the most basic for every branch of mathematics and quantity.

The terms function, mapping, correspondence and transformation usually have the same meaning.

quadratic function

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Generally speaking, there is the following relationship between independent variable x and dependent variable y:

y=ax^2+bx+c

(a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a

Y is called the quadratic function of X.

The right side of a quadratic function expression is usually a quadratic trinomial.

X is an independent variable and y is a function of X.

Three Expressions of Quadratic Function

General formula: y = ax 2+bx+c (a, b and c are constants, and a≠0).

Vertex: y = a(x-h)2+k[ vertex of parabola P(h, k)] For quadratic function y = ax 2+bx+c, its vertex coordinates are (-b/2a, (4ac-b 2)/4a).

Intersection point: y=a(x-x? )(x-x？ ) [only when it is related to the x axis A(x? , 0) and B(x? 0) parabola]

Where x 1, 2 =-b √ b 2-4ac.

Note: Among these three forms of mutual transformation, there are the following relations:

______

h=-b/2a k=(4ac-b^2)/4a x？ ，x？ =(-b √b^2-4ac)/2a

Quadratic function image

The image of quadratic function y = x 2 in plane rectangular coordinate system,

It can be seen that the image of quadratic function is a parabola.

Properties of parabola

1. Parabola is an axisymmetric figure. The symmetry axis is a straight line x = -b/2a.

The only intersection of the symmetry axis and the parabola is the vertex p of the parabola.

Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).

2. The parabola has a vertex p, and the coordinate is P (-b/2a, (4ac-b 2)/4a).

-b/2a=0, p is on the y axis; When δ = b 2-4ac = 0, p is on the x axis.

3. Quadratic coefficient A determines the opening direction and size of parabola.

When a > 0, the parabola opens upward; When a < 0, the parabola opens downward.

The larger the |a|, the smaller the opening of the parabola.

4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.

When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis;

When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.

5. The constant term c determines the intersection of parabola and Y axis.

The parabola intersects the Y axis at (0, c)

6. Number of intersections between parabola and X axis

When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.

When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.

_______

When δ = b 2-4ac < 0, the parabola has no intersection with the X axis. The value of x is an imaginary number (the reciprocal of the value of x =-b √ b 2-4ac, multiplied by the imaginary number I, and the whole formula is divided by 2a).

When a>0, the function obtains the minimum value f (-b/2a) = 4ac-b2/4a at x= -b/2a; In {x | x-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {x | x ≥ 4ac-b 2/4a}, and vice versa.

When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is even, and the analytical expression is transformed into y = ax 2+c (a ≠ 0).

Quadratic function and unary quadratic equation

In particular, the quadratic function (hereinafter called function) y = ax 2+bx+c,

When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).

That is, ax 2+bx+c = 0.

At this point, whether the function image intersects with the X axis means whether the equation has real roots.

The abscissa of the intersection of the function and the X axis is the root of the equation.

1. quadratic function y = ax 2, Y = A (X-H) 2, Y = A (X-H) 2+K, y = ax 2+bx+c (among all kinds, a≠0) has the same image shape, but different positions.

Analytical formula

y=ax^2

y=a(x-h)^2

y=a(x-h)^2+k

y=ax^2+bx+c

Vertex coordinates

(0,0)

(h，0)

(h，k)

(-b/2a,sqrt[4ac-b^2]/4a)

axis of symmetry

x=0

x=h

x=h

x=-b/2a

When h>0, the parabola y = ax 2 is moved to the right by H units in parallel, and the image of y = a (x-h) 2 can be obtained.

When h < 0, it is obtained by moving |h| units in parallel to the left.

When h>0, k>0, the parabola y = ax 2 is moved to the right by H units in parallel, and then moved up by K units, the image of y = a (x-h) 2+k can be obtained;

When h>0, k<0, the parabola y = ax 2 is moved to the right by h units in parallel, and then moved down by | k units, and the image of y = a (x-h) 2+k is obtained;

When h < 0, k >; 0, moving the parabola to the left by |h| units in parallel, and then moving it up by k units to obtain an image with y = a (x-h) 2+k;

When h < 0, k<0, move the parabola to the left by |h| units in parallel, and then move it down by |k| units to obtain an image with y = a (x-h) 2+k;

Therefore, it is very clear to study the image of parabola y = ax 2+bx+c (a ≠ 0) and change the general formula into the form of Y = A (X-H) 2+K through the formula, so as to determine its vertex coordinates, symmetry axis and approximate position of parabola, which provides convenience for drawing images.

2. the image of parabola y = ax 2+bx+c (a ≠ 0): when a >: 0, the opening is upward, when a.

3. parabola y = ax 2+bx+c (a ≠ 0), if a >；; 0, when x ≤ -b/2a, y decreases with the increase of x; When x ≥ -b/2a, y increases with the increase of x, if a

4. The intersection of the image with parabola y = ax 2+bx+c and the coordinate axis:

(1) The image must intersect with the Y axis, and the coordinate of the intersection point is (0, c);

(2) when △ = b 2-4ac >; 0, the image intersects the x axis at two points A(x? , 0) and B(x? 0), where x 1, x2 is the unary quadratic equation ax 2+bx+c = 0.

(a≠0)。 The distance between these two points AB=|x? -x？ In addition, the distance between any pair of symmetrical points on the parabola can be | 2× (-b/2a)-a | (a is one of the points).

When △ = 0, the image has only one intersection with the X axis;

When delta < 0. The image does not intersect with the x axis. When a >; 0, the image falls above the X axis, and when X is an arbitrary real number, there is y >；; 0; When a<0, the image falls below the X axis, and when X is an arbitrary real number, there is Y.

5. the maximum value of parabola y = ax 2+bx+c: if a>0 (a <; 0), then when x= -b/2a, the minimum (large) value of y = (4ac-b 2)/4a.

The abscissa of the vertex is the value of the independent variable when the maximum value is obtained, and the ordinate of the vertex is the value of the maximum value.

6. Find the analytic expression of quadratic function by undetermined coefficient method.

(1) When the given condition is that the known image passes through three known points or three pairs of corresponding values of known x and y, the analytical formula can be set to the general form:

y=ax^2+bx+c(a≠0).

(2) When the given condition is the vertex coordinate or symmetry axis of the known image, the analytical formula can be set as the vertex: y = a (x-h) 2+k (a ≠ 0).

(3) When the given condition is that the coordinates of two intersections between the image and the X axis are known, the analytical formula can be set as two formulas: y=a(x-x? )(x-x？ )(a≠0)。

7. The knowledge of quadratic function can be easily integrated with other knowledge, resulting in more complex synthesis problems. Therefore, the comprehensive question based on quadratic function knowledge is a hot topic in the senior high school entrance examination, which often appears in the form of big questions.

Typical examples of senior high school entrance examination

1. (Xicheng District, Beijing) The symmetry axis of parabola y=x2-2x+ 1 is ().

(a) row x= 1 (B) row x=- 1 (C) row x=2 (D) row x=-2.

Test Center: Symmetry axis of quadratic function Y = AX2+BX+C. 。

Comments: Because the symmetry axis equation of parabola y=ax2+bx+c is: y=-, substituting a= 1 and b=-2 into the known parabola to get x= 1, so option A is correct.

Another method: the parabola formula can be in the form of y=a(x-h)2+k, the symmetry axis is x=h, and the known parabola formula is y=(x- 1)2, so the symmetry axis is x= 1, so A should be chosen.

2. (Dongcheng District, Beijing) has an image of a quadratic function, and three students described some characteristics of it:

A: The symmetry axis is a straight line x = 4；;

B: the abscissa of the two intersections with the X axis is an integer;

C: The ordinate intersecting with the Y axis is also an integer, and the area of the triangle with these three intersections as its vertices is 3.

Please write a quadratic resolution function that satisfies all the above characteristics.

Test center: the solution of quadratic function y=ax2+bx+c

Note: Let the analytical formula be y=a(x-x 1)(x-x2) and x 1 < x2. The two intersections between the image and the X axis are a (x 1, 0) and b (x2, 0) respectively, and the coordinates of the intersections with the Y axis are (0).

The symmetry axis of parabola is the straight line x=4,

∴x2-4=4-x 1, that is, x 1+ x2=8 ①.

∵s△abc=3,∴(x2- x 1)| a x 1 x2 | = 3，

Namely: x2- x 1= ②

① ② Two formulas are added and subtracted: x2=4+, x 1=4-

∵x 1, x2 is an integer, ax 1x2 is also an integer, ∴ax 1x2 is a divisor of 3, * * can be taken as: 1, 3.

When ax 1x2 = 1, x2=7, x 1= 1, and a =

When ax 1x2 = 3, x2=5, x 1=3 and a = 3.

So the analytical formula is: y = (x-7) (x- 1) or y = (x-5) (x-3).

That is, y=x2-x+ 1 or y=-x2+x- 1 or y=x2-x+3 or y=-x2+x-3.

Note: in this question, just fill in an analytical formula or guess and verify. For example, guess that the intersection with the X axis is A (5 5,0) and B (3 3,0). Then find out a from the conditions of the problem and see if c is an integer. If there is, the guess can be verified, just fill it in.

5. (Hebei Province) As shown in figure 13-28, if the image of quadratic function y=x2-4x+3 intersects with the X axis at points A and B, and intersects with the Y axis at point C, the area of △ABC is ().

a、6 B、4 C、3 D、 1

Test site: the image of quadratic function y=ax2+bx+c and the application of its properties.

Comments: From the function image, we can know that the coordinate of point C is (0,3), and then from x2-4x+3=0, we can get x1= kloc-0/,x2=3, so the distance between point A and point B is 2. Then the area of △ABC is 3, so C should be chosen.

Figure 13-28

6. Psychologists in Anhui Province have found that there is a functional relationship between students' ability to accept concepts y and the time to put forward concepts x (unit: minutes): Y =-0. 1x2+2.6x+43 (0 < x < 30). The greater the value of y, the stronger the acceptability.

In what range of (1)x, students' acceptance ability is gradually enhanced? In what range of X, students' acceptance is gradually decreasing?

(2) What is the acceptability of students when the score is 10?

(3) What scores do students accept the most?

Test site: properties of quadratic function y = AX2+BX+C

Comment: parabola y=-0. 1x2+2.6x+43 changed to vertex: y =-0.1(x-13) 2+59.9. According to the properties of parabola, it can be known that the opening is downward. When x≤ 13, y increases with the increase of x, and when 13, y decreases with the increase of x. The range of independent variables of this function is: 0≤x≤30, so the two ranges should be 0 ≤ x ≤13; 13≤x≤30. Substitute x= 10 to find the function value. From the vertex analytic formula, the acceptance ability is the strongest at 13 minutes. The problem solving process is as follows:

Solution: (1) y =-0.1x 2+2.6x+43 =-0.1(x-13) 2+59.9.

Therefore, when 0≤x≤ 13, students' acceptance ability is gradually enhanced.

When 13 < x ≤ 30, students' acceptance ability gradually decreases.

(2) When x= 10, y =-0.1(10-13) 2+59.9 = 59.

When the score is 10, the students' acceptance ability is 59.

(3) When x = 13, y takes the maximum value.

Therefore, in the score of 13, students' acceptance ability is the strongest.

9. A store in Hebei Province sells an aquatic product at a cost of 40 yuan per kilogram. According to market analysis, if it is sold in 50 yuan per kilogram, it can sell 500 kilograms a month; For every increase in the unit sales price of 1 yuan, the monthly sales volume will decrease by 10 kg. Please answer the following questions about the sale of this aquatic product:

(1) When the sales unit price is set to 55 yuan per kilogram, calculate the monthly sales volume and monthly sales profit;

(2) Let the sales unit price be X yuan per kilogram and the monthly sales profit be Y yuan, and find the functional relationship between Y and X (it is not necessary to write the value range of X);

(3) The store wants to make a monthly sales profit of 8,000 yuan when the monthly sales cost does not exceed 1 10,000 yuan. What should the sales unit price be?

Solution: (1) When the sales unit price is set to 55 yuan per kilogram, the monthly sales volume is: 500-(55–50) ×10 = 450 (kg), then the monthly sales profit is

: (55–40) × 450 = 6750 (yuan).

(2) When the sales unit price is X yuan per kilogram, the monthly sales volume is [500-(x–50) ×10] kilograms, and the sales profit per kilogram is (x–40) yuan, then the monthly sales profit is:

y =(x–40)[500-(x–50)× 10]=(x–40)( 1000– 10x)=– 10x 2+ 1400 x-

The resolution function of y and x is: y =–10x2+1400x–40000.

(3) Make the monthly sales profit reach 8000 yuan, that is, y=8000, ∴–10x2+1400x–40000 = 8000,

Namely: x2–140x+4800 = 0,

Solution: x 1=60, X2 = 80.

When the sales unit price is set at one kilogram of 60 yuan, the monthly sales volume is: 500-(60-50)× 10 = 400 (kg), and the monthly sales cost is:

40×400= 16000 (yuan);

When the sales unit price is set at one kilogram of 80 yuan, the monthly sales volume is: 500-(80-50)× 10 = 200 (kg), and the monthly sales unit price cost is:

40×200=8000 (yuan);

Since 8000 < 10000 < 16000, and the monthly sales cost cannot exceed 10000 yuan, the sales unit price should be set at 80 yuan per kilogram.

linear function

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I. Definitions and definitions:

Independent variable x and dependent variable y have the following relationship:

Y=kx+b(k, b is a constant, k≠0)

It is said that y is a linear function of x.

In particular, when b=0, y is a proportional function of x.

Two. Properties of linear functions:

The change value of y is directly proportional to the corresponding change value of x, and the ratio is K.

That is △ y/△ x = K.

Three. Images and properties of linear functions;

1. exercises and graphics: through the following three steps (1) list; (2) tracking points; (3) Connecting lines can make images of linear functions-straight lines. So the image of a function only needs to know two points and connect them into a straight line.

2. Property: any point P(x, y) on the linear function satisfies the equation: y = kx+b.

3. Quadrant where k, b and function images are located.

When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;

When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.

When b > 0, the straight line must pass through the first and second quadrants; When b < 0, the straight line must pass through three or four quadrants.

Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.

At this time, when k > 0, the straight line only passes through one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.

Four. Determine the expression of linear function:

Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..

(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.

(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, two equations can be listed:

Y 1 = KX 1+B 1，Y2 = KX2+B2。

(3) Solve this binary linear equation and get the values of K and B. ..

(4) Finally, the expression of the linear function is obtained.

The application of verb (verb's abbreviation) linear function in life

1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.

2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t, and the original water quantity s in the pool is set. G = S- feet.

inverse proportion function

A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.

The range of the independent variable x is all real numbers that are not equal to 0.

The image of the inverse proportional function is a hyperbola.

As shown in the figure, the function images when k is positive and negative (2 and -2) are given above.

trigonometric function

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Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.

Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.

Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.

It has six basic functions:

Function name: sine cosine tangent cotangent secant cotangent

Symbol sin cos tan cot sec csc

Sine function sin(A)=a/h

Cosine function cos(A)=b/h

Tangent function tan(A)=a/b

Cotangent function cot(A)=b/a

In a certain change process, the two variables X and Y, for each value of X within a certain range, Y has a certain value corresponding to it, and Y is a function of X. This relationship is generally expressed by y=f(x).