Question

函數(shù)f (x)=log₅(x²+1),  x∈[2, +∞的反函數(shù)是             （   ）

A．g (x)=( x≥0)	B．g (x)=( x≥1)
C．g (x)=( x≥0)	D．g (x)=( x≥1)

Answer 1

D

解法一：令y=log₅(x²+1)，可得5^y= x²+1,∴x²=5^y－1, 又∵x∈[2, +∞即x>0.
∴x=.∵x≥2，∴x²+1≥5，y=log₅(x²+1)≥1.
∴函數(shù)f (x)=log₅(x²+1), x∈[2, +∞的反函數(shù)是g (x)=（ x≥1）。故選D.
解法二：∵x≥2，∴x²+1≥5，原函數(shù)y=log₅(x²+1)≥1.由原函數(shù)和反函數(shù)中x，y的對(duì)應(yīng)關(guān)系知反函數(shù)中的x≥1，排除A、 C，而B中 y=>2, 排除B. 故選D.
解法三：原函數(shù)f (x)=log₅(x²+1)經(jīng)過點(diǎn)(2，1)，反函數(shù)y=g (x)經(jīng)過點(diǎn)(1，2)，以 (1，2)點(diǎn)代入排除A、B，又原函數(shù)中y≥1，從而反函數(shù)中x≥1，排除 C，故選D.