y=e^x-e^-x/2的反函数拜托写一下详解

Question

y=e^x-e^-x/2的反函数拜托写一下详解

Answer 1

∵y=(e^x-e^(-x))/2 ==>2y=e^x-e^(-x)
==>2ye^x=e^(2x)-1
==>e^(2x)-2ye^x=1
==>e^(2x)-2ye^x+y²=1+y²
==>(e^x-y)²=1+y²
==>e^x-y=±√(1+y²)
==>e^x=y±√(1+y²)
==>x=ln│y±√(1+y²)│
==>x=±ln│y+√(1+y²)│ (∵ln│y-√(1+y²)│=-ln│y+√(1+y²)│)
==>x=±ln(y+√(1+y²)) (∵y+√(1+y²)>0)
∴原函数的反函数是 x=±ln(y+√(1+y²)).