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虽然在中学课本中我们只学习了有限的几种初等函数,但通过它们的复合、四则运算可以构造出许多新的函数.这里笔者将对形如y=ax~(2n)+bx~n+c(其中a≠0,n∈N)的函数的性质进行初步探讨. 显然F(x)=ax~(2n)+bx~n+c(其中a≠0,n∈N)是一类多项式函数,它的定义域为R,是由y=f(u)=au2+bu+c和u=x~(n∈N)复合而成.利用复合函数的单调性法则,即“同调得增,异调得减”,若能画出其图像草图,则其性质就一目了然.
Although we only studied a limited number of elementary functions in the middle school textbooks, many new functions can be constructed through their compound and four operations. Here I will describe the shape of y=ax~(2n)+bx~n+. The properties of the function of c (where a ≠ 0, n ∈ N) are preliminary discussed. Obviously F(x) = ax ~ (2n) + bx ~ n + c (where a ≠ 0, n ∈ N) is a class of polynomials. The function, whose domain is defined by R, is a compound of y=f(u)=au2+bu+c and u=x~(n∈N). It uses the monotonicity law of the compound function, ie , if you draw a sketch of the image, its nature is clear.