The advantages and disadvantages of the three representation methods of functions

The advantages and disadvantages of the three methods of expressing functions are as follows:

The three methods of expressing functions: image method, list method, and analytical method.

The list method can directly see the quantitative relationship between the dependent variable and the independent variable, but has the disadvantage of being unintuitive.

The image method can be seen intuitively, and the changing trend of the function as the independent variable changes. The disadvantage is that the numerical value cannot be seen.

The analytical method is convenient for studying the properties of functions, but has the disadvantage of being too abstract.

Boundedness of function:

Suppose the domain of function f(x) is D, and the number set X is included in D. If there is a number K1 such that f(x)≤K1 holds for any x∈X, then the function f(x) is said to have an upper bound on X, and K1 is called a function f(x) on X. boundary. If there is a number K2 such that f(x)≥K2 is true for any x∈X, then the function f(x) is said to have a lower bound on X, and K2 is called a lower bound of the function f(x) on X.

If there is a positive number M such that |f(x)|≤M is true for any x∈X, then the function f(x) is said to be bounded on X. If such M does not exist , it is said that the function f(x) is unbounded on X.

The necessary and sufficient condition for the function f(x) to be bounded on X is that it has both an upper bound and a lower bound on X.

Monotonicity of function:

Suppose the domain of function f(x) is D, and the interval I is included in D. If for any two points x1 and x2 on the interval I, when x1f(x2), then the function f(x) is said to be monotonically decreasing in the interval I. Functions that increase and decrease monotonically are collectively called monotonic functions.

Periodicity of function:

Suppose the domain of function f(x) is D. If there is a positive number l such that for any x∈D there is (x±l)∈D, and f(x+l)=f(x) is always true, then f(x) is called a periodic function, and l is called is the period of f(x). Usually we say that the period of a periodic function refers to the minimum positive period. The domain D of a periodic function is an unbounded interval with at least one side. If D is bounded, the function is not periodic.

Not every periodic function has a minimum positive period, such as the Dirichlet function.