The general form of logarithmic function is that it is actually the inverse of exponential function. Therefore, the stipulation of a in exponential function is also applicable to logarithmic function.

The figure on the right shows the function diagram of different size A:

You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.

The domain of (1) logarithmic function is a set of real numbers greater than 0.

(2) The range of logarithmic function is the set of all real numbers.

(3) The function always passes (1, 0).

(4) When a is greater than 1, it is monotone increasing function and convex; When a is less than 1 and greater than 0, the function is monotonically decreasing and concave.

(5) Obviously, the logarithmic function is unbounded.

exponential function

The general form of exponential function is that from the above discussion of power function, we can know that if X can take the whole set of real numbers as the domain, then it only needs to be done.

As shown in the figure, different sizes of a will affect the function diagram.

You can see:

The domain of (1) exponential function is the set of all real numbers, where a is greater than 0. If a is not greater than 0, there will be no continuous interval in the definition domain of the function, so we will not consider it.

(2) The range of exponential function is a set of real numbers greater than 0.

(3) The function graph is concave.

(4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.

(5) We can see an obvious law, that is, when a tends to infinity from 0 (of course, it can't be equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing.

(6) Functions always infinitely tend to a certain direction on the X axis and never intersect.

(7) The function always passes (0, 1).

Obviously the exponential function is unbounded.

odevity

Note: (1) is an odd function; (2) is an even function.

1. Definition

Generally, for the function f(x)

(1) If any x in the function definition domain has f (-x) =-f(x), then the function f(x) is called odd function.

(2) If any x in the function definition domain has f(-x)=f(x), the function f(x) is called an even function.

(3) If f(-x)=-f(x) and f(-x)=f(x) are true for any x in the function definition domain, then the function f(x) is both a odd function and an even function, which is called an even-even function.

(4) If f(-x)=-f(x) and f(-x)=f(x) cannot be established for any x in the function definition domain, then the function F (x) is neither a odd function nor an even function, which is called an even-even function.

Description: ① Odd and even are global properties of functions, and they are global.

② The domains of odd and even functions must be symmetrical about the origin. If the domain of a function is not symmetric about the origin, then the function must not be an odd (or even) function.

(Analysis: To judge the parity of a function, we should first check whether its domain is symmetrical about the origin, then simplify and sort it out strictly according to the definitions of odd and even, and then compare it with f(x) to draw a conclusion. )

③ The basis for judging or proving whether a function has parity is definition.

2. The characteristics of the odd-even function image:

Theorem The image of odd-numbered function is a centrally symmetric figure about the origin, and the image of even-numbered function is a figure about the Y axis or axis symmetry.

F(x) is odd function's image "= =" F (x) is symmetrical about the origin.

Point (x, y)→(-x, -y)

Odd function monotonically increases in a certain interval and monotonically increases in its symmetric interval.

Even function monotonically increases in a certain interval, but monotonically decreases in its symmetric interval.

3.

Parity function operation

(1). The sum of two even functions is an even function.

(2) The sum of the two odd function is odd function.

(3) The sum of an even function and a odd function is a non-odd function and non-even function.

(4) The product obtained by multiplying two even functions is an even function.

(5) The product obtained by multiplying two odd function is an even function.

(6) The product of even function multiplied by odd function is odd function.

Domain of definition

Let A and B be two sets of nonempty numbers. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A-B is called a function from set A to set B, denoted as Y = F (X), and X belongs to set A ... where X is called an independent variable, and the value range A of X is called the definition domain of the function;

range

Name definition

In a function, the range of the dependent variable is called the range of the function, which is the set of all the values of the dependent variable in the definition domain in mathematics.

Common methods for evaluating domains

(1) reduction method; (2) Image method (combination of numbers and shapes),

(3) Monotonicity of the function,

(4) matching method, (5) substitution method, (6) inverse function method, (7) discriminant method, (8) compound function method, (9) trigonometric substitution method, (10) basic inequality method, etc.

Misunderstanding of Function Value Domain

Definition domain, corresponding rules and value domain are the three basic "components" of function construction. There is no doubt that the principle of "domain priority" is implemented in mathematics at ordinary times. However, everything has duality, and while strengthening the problem of domain, it is often weakened or discussed. The exploration of the range problem leads to "hard" and "soft" in one hand, which makes students' mastery of functions intermittent. In fact, the positions of the definition domain and the value domain are equivalent, so they must not be too detailed, not to mention that they are always in mutual transformation (a typical example is the definition domain and the value domain of reciprocal functions). If the range of a function is infinite, it is not always easy to find the range of the function. The operational nature of dependency inequality is sometimes invalid, and the function value must be considered in combination with the parity, monotonicity, boundedness and periodicity of the function. In order to get the correct answer, from this point of view, it is sometimes more difficult to evaluate the domain than to find the domain. Practice has proved that strengthening the research and discussion on the method of finding the domain is helpful for us to understand the function in the domain, thus deepening our understanding of the essence of the function.

Is "scope" the same as "scope"?

"Value range" and "value range" are two concepts that we often encounter in our study, and many students often confuse them. Actually, they are two different concepts. "range" is a set of all function values (that is, every element in the set is the value of this function), while "Range" is only a set of some values that meet a certain condition (that is, all elements in the set may not meet this condition). That is to say, the scope is the scope, but the scope is not necessarily the scope.