Question

已知函數(shù)f(x)=.

(1)證明函數(shù)f(x)在定義域上有反函數(shù),并求出反函數(shù);

(2)反函數(shù)的圖象與直線y=x有無(wú)交點(diǎn)?

Answer 1

(1)證明:∵f(x)的定義域?yàn)檎龑?shí)數(shù)集,

∴當(dāng)0＜x₁＜x₂時(shí),f(x₁)-f(x₂)=()-()=()(1+)＜0.

∴f(x₁)＜f(x₂),即f(x)在(0,+∞)上是增函數(shù).

∴f(x)有反函數(shù).當(dāng)x∈(0,+∞)時(shí),∈(-∞,+∞).∴反函數(shù)的定義域?yàn)镽.

由y=,得x-y-1=0.

解得=.∵y＜,

∴y-＜0.而＞0,

∴=,x=(y+)².

∴f^-1(x)=(x+)²(x∈R).

(2)解：y=f^-1(x)與y=f(x)的圖象關(guān)于y=x對(duì)稱(chēng),故只需判斷y=f(x)與y=x有無(wú)交點(diǎn).

由得x=.

∴x(1-)=1.

當(dāng)0＜x＜1時(shí),0＜1-＜1.∴0＜x(1-)＜1,此時(shí)方程無(wú)實(shí)數(shù)根.

當(dāng)x≥1時(shí),x(1-)≤0,方程無(wú)實(shí)根.

∴y=f(x)與y=x無(wú)交點(diǎn).

從而y=f^-1(x)與y=x無(wú)交點(diǎn).