Mathematics

Problem

Let

f: \mathbb R \to (-1,\infty)

be defined by

f(x)=e^{2x}-1

. (i) Show that

f

is bijective. (ii) Obtain a function

g:(-1,\infty)\to\mathbb R

such that

(g\circ f)(x)=x

for all

x\in\mathbb R

. (iii) Verify that

(f\circ g)(y)=y

for all

y>-1

. (iv) State the domains and ranges of

f

and

g

that make the above compositions valid, and conclude that

f

and

g

are inverses of each other.