welcome

The definition of the function has been described in the phase. After entering high school, I learned mapping on the basis of learning set, and then I learned the concept of function, mainly to clarify the function from the perspective of mapping. At this time, I can use the functions that students already know, especially quadratic functions, as an example to understand the concept of functions more deeply. Quadratic function is the mapping "f: a → b" from one set A (domain) to another set B (domain), so that the element Y in set B = AX2+BX+C (A ≠ 0). The element X corresponding to set A is marked as f(x)=ax2+bx+c(a≠0), where ax2+bx+c represents the corresponding rule of element X in the definition domain and the image in the value domain, so that students can have a clear understanding of the concept of function. After students have mastered the mark of function value, they can further deal with the following problems:

Formula I: Given that f(x)=2x2+x+2, find f(x+ 1).

F(x+ 1) cannot be understood as the function value when x=x+ 1, but only as the function value with x+ 1 as the independent variable.

Type ⅱ: let f(x+ 1)=x2-4x+ 1 and find f(x).

This problem can be understood as knowing the corresponding rule F, and the image of the element x+ 1 in the domain is x2-4x+ 1. The essence of finding the image of the element X in the domain is to find the corresponding rule.

Generally speaking, there are two methods:

(1) Represents the given expression as a polynomial of x+ 1

F (x+ 1) = x2-4x+1= (x+1) 2-6 (x+1)+6, and then replace x+1with x to get f (x) = x2.

(2) Variable replacement. It has strong adaptability and can be applied to general functions.

Let t=x+ 1, then x=t- 1.

∴ f (t) = (t-1) 2-4 (t-1)+1= t2-6t+6, so that f(x)=x2-6x+6.

Monotonicity, Maximum and Image of Quadratic Function