Question

設(shè)函數(shù)f(x)=+lg,

(1)求函數(shù)f(x)的定義域;

(2)判斷函數(shù)f(x)的單調(diào)性,并給出證明;

(3)已知函數(shù)f(x)的反函數(shù)f ^-1(x),問函數(shù)y=f ^-1(x)的圖象與x軸有交點(diǎn)嗎?若有,求出交點(diǎn)坐標(biāo);若無交點(diǎn),說明理由.

Answer 1

解析:(1)由3x+5≠0且＞0,解得x≠-且-＜x＜.取交集得-＜x＜.?

(2)令μ(x)=3x+5,隨著x增大,函數(shù)值減小,所以在定義域內(nèi)是減函數(shù);?

=-1+隨著x增大,函數(shù)值減小,所以在定義域內(nèi)是減函數(shù).

又y=lgu在定義域內(nèi)是增函數(shù),根據(jù)復(fù)合單調(diào)性可知,y=lg是減函數(shù),所以f(x)=+lg是減函數(shù).

(3)因?yàn)橹苯忧骹(x)的反函數(shù)非常復(fù)雜且不易求出,于是利用函數(shù)與其反函數(shù)之間定義域與值域的關(guān)系求解.?

設(shè)函數(shù)f(x)的反函數(shù)f ^-1(x)與x軸的交點(diǎn)為(x ₀,0).根據(jù)函數(shù)與反函數(shù)之間定義域與值域的關(guān)系可知,f(x)與y軸的交點(diǎn)是(0,x ₀),將(0,x ₀)代入f(x),解得x ₀=.所以函數(shù)y=f ^-1(x)的圖象與x軸有交點(diǎn),交點(diǎn)為(,0).