Question

数学题求反函数1.1.6如图所示,

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)).