The concepts of 1, correspondence, mapping and function are both different. Mapping is a special correspondence, and function is a special mapping.

2, for the concept of function, should pay attention to the following points:

(1) Knowing the three elements of a function can determine whether two functions are the same.

(2) Master three representations-list method, analytical method and mirror method, and seek the functional relationship between variables according to practical problems, especially the analytical formula of piecewise function.

(3) If y=f(u) and u=g(x), then y=f[g(x)] is called a composite function of f and g, where g(x) is an inner function and f(u) is an outer function.

3. The general steps to find the inverse function of the function y=f(x):

(1) Determine the range of the original function, that is, the definition range of the inverse function;

(2) x = f-1(y) is obtained from the analytical formula of y=f(x);

(3) Exchange x and y to get the idiomatic expression y = f- 1 (x) of the inverse function, and mark the domain.

Note ①: For the inverse function of piecewise function, first find the inverse function on each segment separately, and then merge them together.

② Be familiar with the application, find the value of F- 1 (x0), and make rational use of this conclusion to avoid the process of finding the inverse function, thus simplifying the operation.

(2), the analytical formula and definition of the function

1, the function and its domain are an inseparable whole, and the function without domain does not exist. Therefore, in order to correctly write the analytical expression of the function, we must find the corresponding law between variables and the definition domain of the function. There are usually three ways to find the function domain:

(1) Sometimes a function comes from a practical problem, so the independent variable x has practical significance, and the domain should be considered in combination with the practical significance;

(2) Find the domain of the analytic formula of the known function, as long as the analytic formula is meaningful, such as:

The denominator of (1) score must not be zero;

(2) The number of even roots is not less than zero;

③ The true value of logarithmic function must be greater than zero;

④ The bases of exponential function and logarithmic function must be greater than zero and not equal to1;

⑤ tangent function y=tanx(x∈R and k∈Z) and cotangent function y=cotx(x∈R, x ? k π, k∈Z) in trigonometric functions.

It should be noted that when the analytic expression of a function consists of several parts, the domain is the common part (i.e. intersection) of each meaningful independent variable.

(3) Knowing the domain of one function and finding the domain of another function mainly consider the profound meaning of the domain.

It is known that the domain of f(x) is [a, b], the domain of f[g(x)] means that x satisfies the range of a≤g(x)≤b, and the domain of f[g(x)] is known as x ∈ [a, b].

2. Generally speaking, there are four ways to find the analytic expression of a function.

(1) When it is necessary to establish a function relationship according to practical problems, it is necessary to introduce appropriate variables and find the analytical formula of the function according to the relevant knowledge of mathematics.

(2) Sometimes, given the characteristics of a function, we can use the undetermined coefficient method to find the analytical expression of the function. For example, if the function is linear, let f (x) = ax+b (a ≠ 0), where a and b are undetermined coefficients. According to the conditions of the problem, list the equations and find out a and B.

(3) If the expression of the compound function f[g(x)] is given, method of substitution can be used to find the expression of the function f(x), and then the range of g(x) must be found, which is equivalent to finding the domain of the function.

(4) If it is known that f(x) satisfies an equation, and other unknowns (such as F (-x), etc. ) Except for f(x), all other equations appearing in this equation must be constructed according to the known equation, and the expression of f(x) can be obtained by solving the equation.

(3) The range and maximum value of the function

The range of 1. function depends on the defined range and the corresponding law. No matter what method is used to find the function range, we should first consider defining the range. The common methods to find the range of functions are as follows:

(1) direct method: also known as observation method, for functions with simple structure, the range of the function can be directly observed by applying the properties of inequality to the analytical expression of the function.

(2) Substitution method: A given complex variable function is transformed into another simple function re-evaluation domain by algebraic or trigonometric substitution. If the resolution function contains a radical, algebraic substitution is used when the radical is linear and trigonometric substitution is used when the radical is quadratic.

(3) Inverse function method: By using the relationship between the definition domain and the value domain of the function f(x) and its inverse function f- 1 (x), the value domain of the original function can be obtained by solving the definition domain of the inverse function, and the function value domain with the shape of (a≠0) can be obtained by this method.

(4) Matching method: For the range problem of quadratic function or function related to quadratic function, the matching method can be considered.

(5) Evaluation range of inequality method: Using the basic inequality A+B ≥ [A, b∈(0, +∞)], we can find the range of some functions, but we should pay attention to the condition of "one positive, two definite, three phases, etc." Sometimes you need skills such as Fang.

(6) Discriminant method: y=f(x) is transformed into a quadratic equation about x, and the definition domain is evaluated by "△≥0". The characteristic of the question type is that the analytical formula contains roots or fractions.

(7) Finding the domain by using the monotonicity of the function: When the monotonicity of the function on its domain (or a subset of the domain) can be determined, the range of the function can be found by using the monotonicity method.

(8) Finding the range of function by combining numbers and shapes: using the geometric meaning expressed by the function, finding the range of function by geometric methods or images, that is, finding the range of function by combining numbers and shapes.

2. Find the difference and connection between the maximum value of the function and the range.

The common method of finding the maximum value of a function is basically the same as the method of finding the function value domain. In fact, if there is a minimum (maximum) number in the range of a function, this number is the minimum (maximum) value of the function. Therefore, the essence of finding the maximum value of a function is the same as that of the evaluation domain, but the angle of asking questions is different, so the way of answering questions is different.

For example, the value range of the function is (0, 16), the maximum value is 16, and there is no minimum value. For example, the range of the function is (-∞, -2]∩[2,+∞), but this function has no maximum and minimum, only after changing the definition of the function, such as X >；; 0, and the minimum value of the function is 2. The influence of definition domain on the range or maximum value of a function can be seen.

3. The application of maximum function in practical problems.

The application of function maximum is mainly reflected in solving practical problems with function knowledge. Many practical problems are commonly expressed in words such as "lowest project cost", "maximum profit" or "maximum (minimum) area (volume)". When solving, we should pay special attention to the restriction of practical significance on independent variables, so as to get the maximum value correctly.

(4) Parity of functions

1. Definition of function parity: For function f(x), if any x in the function definition domain has f (-x) =-f(x) (or f (-x) = f (x)), then function f(x) is called odd function (or even function).

To correctly understand the definitions of odd function and even functions, we should pay attention to two points: the symmetry of the domain on the (1) number axis is a necessary and sufficient condition for the function f(x) to be a odd function or even function; (2) f (x) =-f (x) or f (-x) = f (x) is a unit element in the domain. (Parity is a global property over the domain of a function. ) 。

2. The definition of parity function is the main basis for judging the parity of function. In order to judge the parity of a function, it is sometimes necessary to simplify the equivalent form of the function or application definition:

Pay attention to the application of the following conclusions:

(1) f(|x|) is always an even function whether f(x) is a odd function or an even function;

(2)f(x) and g(x) are odd function in the fields D 1 and D2, respectively, so on D 1∩D2, F (x)+G (x) is odd function, F (x) G (x) is an even function, and similarly ".

(3) The parity of the compound function of the even-odd function is usually an even function;

(4) The derivative function of odd function is even function, and the derivative function of even function is odd function.

3. Some properties and conclusions about parity.

The necessary and sufficient condition for (1) function to be odd function is that its image is symmetric about the origin; The necessary and sufficient condition for a function to be an even function is that its image is symmetric about y 。

(2) If the domain of a function is symmetric about the origin and the value of the function is always zero, then it is both a odd function and an even function.

(3) If odd function f(x) is meaningful at x=0, then f(0)=0 holds.

(4) If f(x) is an interval monotone function with parity, the monotonicity of odd (even) functions in positive and negative symmetric intervals is the same (inverse).

(5) If the domain of f(x) is symmetric about the origin, then f (x) = f (x)+f (-x) is an even function, and g (x) = f (x)-f (-x) is odd function.

(6) Universal parity

If the function y=f(x) has f (a+x) = f (a-x) for any x in the definition domain, then the image of y=f(x) is symmetric about the straight line x=a, that is, y = f (a+x) is an even function. The function y=f(x) applies to any x in the domain.

(5) Monotonicity of functions

1, monotone function

For the function f(x), x 1, x2 defined at any two points on the interval [a, b], when x 1 >; X2, with inequality f (x 1) >: (or

For the understanding of the definition of function monotonicity, we should pay attention to the following three points:

(1) monotonicity is a concept closely related to interval. A function can have different monotonicity in different intervals.

(2) Monotonicity is the "global" property of a function in a certain interval, so the definitions of x 1 and x2 are arbitrary and cannot be replaced by special values.

(3) Monotonicity interval is a subset of the domain, and the discussion of monotonicity must be within the domain.

(4) Pay attention to two equivalent forms of definition:

Let x 1 and x2∈[a, b], then:

(1) About [a, b] is an increasing function;

It is the subtraction function on [a, b].

② On [a, b] is increasing function.

It is the subtraction function on [a, b].

It should be pointed out that: ① Geometrically, the slopes of any two points (x 1, f(x 1) and (x2, f(x2)) on the increase (decrease) function image are all greater than (or less than) zero.

(5) Since definitions are all necessary and sufficient propositions, f(x) is an increasing (decreasing) function, and (or x1>; X2), which shows that monotonicity makes the inequality between independent variables and the inequality between function values "push forward and then close".

5. Monotonicity of compound function y=f[g(x)]

If the monotonicity of u=g(x) in the interval [a, b] is the same as that of y=f(u) in [g(a), g(b)] (or g(b), g(a)), then the composite function y=f[g(x)] in [decreases monotonically.

When studying the monotonicity of a function, it is often necessary to simplify the function first and turn it into discussing the monotonicity of some well-known functions. Therefore, mastering and remembering the monotonicity of linear function, quadratic function, exponential function and logarithmic function will greatly shorten our judgment process.

6. Proof method of monotonicity of function

It is proved according to the definition (1). The steps are as follows: ① Take x 1, x2∈M, X 1

(2) Let the function y=f(x) be derivable in a certain interval.

if f '(x)>； 0, then f(x) is increasing function; If f' (x)

(6), the function of image

The image of function is the intuitive embodiment of function, so we should strengthen the cultivation of drawing, recognizing and using pictures, and cultivate the consciousness of combining numbers with shapes to solve problems.

Find the function expression of the image

Relationship with f(x)

Image transformation from f(x)

y = f(x)b(b & gt； 0)

Translate b units along the y axis.

y = f(x a)(a & gt； 0)

Translate one unit along the x axis.

y=-f(x)

Make a symmetrical figure about the X axis.

y=f(|x|)

The right side is fixed and symmetrical about Y.

y=|f(x)|

The upper part is fixed and the lower part is folded along the X axis.

y=f- 1(x)

Make a symmetrical figure about the straight line y = x.

y = f(ax)(a & gt； 0)

The abscissa is shortened to the original, and the ordinate is unchanged.

y=af(x)

The vertical axis is extended to |a| times of the original, and the horizontal axis remains unchanged.

y=f(-x)

Make a graph about y axis symmetry.

For example, the function f(x) defined on a real number set has f (x+y)+f (x-y) = 2f (x) f (y) for any x, y∈R and f (0) ≠ 0.

① Verification: f (0) =1;

② Prove that y=f(x) is an even function;

(3) If there is a constant c, F (x+c) =-F (x) holds for any x∈R; Find whether the function f(x) is a periodic function, and if so, find a period of it; If not, please explain why.

Train of thought analysis: We call the function without analytic expression abstract function, and generally use the method of assignment to solve this kind of problem.

Answer: ① Let x=y=0, then 2f(0)=2f2(0), because f(0)≠0, so f (0) = 1.

② If x=0, there is f(x)+f (-y) = 2f (0) f (y) = 2f (y), so f (-y) = f (y), which shows that f(x) is an even function.

(3) replace x and y with (c > 0) respectively, and use f (x+c)+f (x) =

So, so f (x+c) =-f (x).

The conclusion in the application of both parties is that f (x+2c) =-f (x+c) =-[-f (x)] = f (x),

So f(x) is a periodic function, and 2c is its period.

Geometric Theorem Menelaus Theorem

A straight line cuts three sides BC, CA and AB of △ABC or its extension lines in D, E and F. ..

Inverse theorem: a straight line cuts the three sides BC, CA, AB or their extension lines of △ABC at D, E and F. If so, then the three points of D, E and F are connected.

Cheval theorem

Take any point o in △ABC, and lines AO, BO and CO intersect at the opposite sides of d, e and f respectively, then = 1.

Inverse theorem: take points d, e and f on the sides BC, CA and AB of △ABC respectively. If = 1, then the straight lines AD, BE and CF intersect at the same point.

ptolemy's theorem

If ABCD is a quadrilateral inscribed in any circle, then.

Inverse theorem: If the quadrilateral ABCD is satisfied, then A, B, C and D are four-point * * * circles.

Siemsen line

If any point on the circumscribed circle of the triangle is different from the vertex of the triangle as the vertical line of the three sides, the straight line of * * * will be vertical. This line is usually called the siemsen Line. The inverse theorem of siemsen's theorem is: If a point is a projective straight line on the straight line where the three sides of a triangle are located, it is on the circumscribed circle of the triangle.

The relevant results are as follows:

(1) calls the vertical center of the triangle. The intersection of h. Simson line and PH is the midpoint of the line segment PH, which is on the nine-point circle. (2) The intersection angle of two Simson lines is equal to the circumferential angle of two points.

(3) If the circumscribed circles of two triangles are the same, a point P on the circumscribed circle corresponds to the intersection angle of the two triangles' Simpson lines, regardless of the position of P. ..

(4) The necessary and sufficient condition for drawing a vertical line from a point to three sides of a triangle is that the point falls on the circumscribed circle of the triangle.

Stewart theorem

Let a point D between △ABC and its bottom two points B and C be known, then there is AB2 DC+AC2 BD-Ad2 BC = BC DC BD.

Triangular auxiliary center

1, the center of the tangent circle is called the edge center of the triangle.

2. The circle tangent to the extension line of one side and the other two sides of a triangle is called the tangent circle of the triangle.

fermat point

In a triangle, the point with the smallest sum of the distances to the three vertices is called the fermat point of the triangle.

(1) If all three internal angles of the triangle ABC are less than 120, then the three distance lines just bisect the rounded corner where fermat point is located. So the fermat point of a triangle is also called the isocenter of a triangle.

(2) If the internal angle of a triangle is not less than 120 degrees, the vertex of this obtuse angle is the point with the smallest distance sum.

Determine (1) For any triangle △ABC, if there is a little E in or on the triangle and EA+EB+EC has the minimum value, then E is fermat point. Fermat point's calculation

(2) If the interior angle of a triangle is greater than or equal to 120, the vertex of this interior angle is the fermat point; If all three internal angles are less than 120, then three points with an opening angle of 120 inside the triangle are the fermat point of the triangle.

Nine-point circle: a nine-point * * * circle with the midpoint of three sides of a triangle, three vertical feet and three Euler points (the midpoint of three line segments obtained by connecting the vertices and vertical centers of the triangle). This circle is usually called a nine-point circle.

Euler Line: The outer center, center of gravity, center of nine points and vertical center of a triangle are located on the same straight line in turn. This straight line is called the Euler line of triangle.

Geometric inequality

Ptolemaic inequality 1: Any convex quadrilateral ABCD must have AC BD ≤ ABCD+AD BC, if and only if ABCD has four points * * * circle, it is an equal sign.

Erdos-Mo Deer inequality: let p be any point in Δ ABC, and the distances from p to BC, CA and AB of Δ ABC are PD=p, PE=q, PF=r, Pa = X, Pb = Y and PC = Z, respectively. Then x+y+z≥2(p+q+r)

3 Waisenbick inequality: Let the length of three sides of △ABC be A, B, C and the area be S, then a2+b2+c2≥4.

Euler inequality: Let the radii of the circumscribed circle and inscribed circle of △ABC be r and r respectively, then R≥2r, and take the equal sign if and only if △ABC is a regular triangle.

The power of circle

Suppose there is a point P and a circle O on the plane, and its radius is R, then OP 2-R 2 is the power of point P to circle O; It can be seen that the power outside the circle is positive, the power inside the circle is negative, and the power on the circle is 0;

radical axis

If 1 gives two circles with different centers on the plane, then the set of idempotent points of these two circles is a straight line, which is called the root axis of these two circles.

Another angle can also be called the locus of idempotent points with two different concentric circles as the root axis.

Correlation theorem

1, the root axes of any two circles on the plane are perpendicular to their connecting lines;

2. If two circles intersect, the root axis of the two circles is the straight line where the chord is located;

3. If two circles are tangent, the root axes of the two circles are their internal common tangents;

4. Mongolian theorem (root center theorem): Any three circles on a plane whose centers are not on the straight line are called their root centers, and their root axes are either parallel to each other or intersect at one point;