y fX exex2 is called hyperbola sine A what is domain of f

y = f(X) = e^x-e^x/2 is called hyperbola sine

(A) what is domain of f ?

(B) show that f is one to one function

(C) show that f^-1(X) =ln(x+underoot of x²+1)

(d) what is range of f ?

Solution

hyperbola sine is y =(e^x-e^-x)/2

a) domain is (- ,)

b) f(a)=f(b)

(e^a-e^-a)/2=(e^b-e^-b)/2

(e^a-e^-a)=(e^b-e^-b)

=>a =b

so f is one to one function

c)

(e^x-e^-x)/2 =y

e^x-e^-x=2y

e^x-(1/e^x)=2y

((e^x)²-1)/e^x=2y

(e^x)²-1=2ye^x

(e^x)²-2ye^x-1=0

let e^x=p

p²-2yp-1=0

by quadratic formula

p =[2y +[(2y)² -4(1)(-1)]]/(2*1)

p =[2y +[4y²+4]]/(2)

p =[2y +2[y²+1]]/(2)

p =y +[y²+1]

e^x=y +[y²+1]

x =ln(y +[y²+1])

f^-1(y)=ln(y +[y²+1])

f^-1(x)=ln(x +[x²+1])

d) range of f = (- ,)