Question

已知函數(shù)y=f(x)的圖象與函數(shù)y=a^x(a＞0且a≠1)的圖象關(guān)于直線y=x對(duì)稱，記g(x)=f(x)［f(x)+f(2)-1］.若y=g(x)在區(qū)間［，2］上是增函數(shù)，則實(shí)數(shù)a的取值范圍是(    )

A.［2，+∞)            B.(0,1)∪(1,2)            C.［,1)            D.(0,］

Answer 1

解析：∵f(x)=log_ax，∴g(x)=log_ax·(log_ax+log_a2-1)=(log_ax)²+(log_a2-1)·log_ax.

    取特值a=，

g(x)=(x)²+(2-1)·x

=(log₂x)²+(-2)·(-log₂x)

=x+2log₂x,                                    ①

    令log₂x=t，則①變?yōu)閠²+2t，

    當(dāng)x∈［,2］時(shí)，t∈［-1,1］,

    此時(shí)t↗，函數(shù)g(x)的對(duì)稱軸為x=-1,∴二次函數(shù)t²+2t↗.∴整個(gè)函數(shù)↗，滿足題意.

    同理，再取特值a=、a=，排除B、C，可知答案選D.

答案：D