已知x≠5/4,求函数f(x)=4x-2+1/(4x-5)的值域

问题描述:

已知x≠5/4,求函数f(x)=4x-2+1/(4x-5)的值域

f(x)=4x-2+1/(4x-5)=(4x-5)+1/(4x-5) +3
若4x-5<0,则f(x)=-[-(4x-5)+1/-(4x-5)] +3≤-2√1+3=1,
仅当-(4x-5)=1/-(4x-5),即x=1时,等号成立;
若4x-5>0,则f(x)=[(4x-5)+1/(4x-5)] +3≥2√1+3=5,
仅当(4x-5)=1/(4x-5),即x=3/2时,等号成立;
即函数f(x)=4x-2+1/(4x-5)的值域为(-∞,1]U[5,+∞).