If we know a problem about the function f(x) and prove the relationship between the properties of f(x) and the satisfaction of f(x), the problem is completed, but we don't know the specific analytical formula of f(x), which is an abstract function problem.

Generally speaking, abstract function refers to a function that does not give (directly or indirectly) specific analytical expressions, but only gives some function symbols and functions that meet certain conditions.

To solve the problem of abstract function, we can adopt many methods, such as function nature, specialization, model function, associative analogy transformation, combination of numbers and shapes, etc.

(1) function property method.

The characteristics of a function are reflected by its properties (such as monotonicity, parity, periodicity, special points, etc.). ), so do abstract functions. We can comprehensively apply the above properties, including solving abstract function problems with the help of special point Boolean equations.

(2) Specialized methods.

Special methods are also called special methods. In order to achieve our expected purpose, the known conditions are appropriately transformed, including the overall transformation of the formula and the substitution of specific numbers. For example, when studying the properties of functions, x is generally replaced by -x or other algebraic expressions; When evaluating, the special values 0, 1 and-1 are often replaced by assignment method.

(3) Model function method.

Model function plays an important role in solving abstract function problems. On the one hand, specific model functions can be used to solve objective problems such as multiple-choice questions and fill-in-the-blank questions. On the other hand, we can use "special case inquiry" to connect specific model functions with analogy and conjecture, and provide ideas and methods for solving subjective problems such as solving problems. Generally, there are the following types of abstract functions:

① Satisfy the relationship.

f(x+y)=f(x)+f(y) (ⅰ)

F(x) is a linear abstract function, and its model function is the proportional function f(x)=kx(k≠0).

Actually f(x+y)=k(x+y)=kx+ky=f(x)+f(y).

Let x=y=0 and f(0)=0, so the image of f(x) must pass through the origin.

Let y=-x and get 0=f(0)=f(x)+f(-x), that is, f(-x)=-f(x), so f(x) is odd function.

Proposition (I) can be generalized as f(x+y)=f(x)+f(y)+b(b is a constant), and its model function is linear function f(x)=kx-b(k≠0).

② Satisfy the relationship.

f(x+y)=f(x) f(y) (ⅱ)

F(x) is an exponential abstract function, and its model function is exponential function f (x) = ax (a >; 0，a≠ 1)。

Actually, f (x+y) = ax+y = ax ay = f (x) f (y).

Let x=y=0 and f(0)= 1, so the curve f(x) must pass through the point (0, 1).

Proposition (Ⅱ) is equivalent to f(x-y)=.

③ Satisfy the relationship.

f(xy)=f(x)+f(y) (x，y∈R+) (ⅲ)

F(x) is a logarithmic abstract function, and its model function is logarithmic function f (x) = logax (a >; 0，a≠ 1)。

Let x=y= 1 and get f( 1)=0, so the curve f(x) must pass through the point (1, 0).

Proposition (Ⅲ) is equivalent to f( )=f(x)-f(y) (x, y∈R+).

④ Satisfy the relational expression.

f(xy)=f(x) f(y)

F(x) is a power abstract function, and its model function is power function f(x)=xn.

For more function questions and answers, please refer to "Lecture on Function in Senior High School" or contact 133@qq.com at 2836395.

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