Mathematics

Problem

Let

f: \mathbb{R}\to(0,\infty)

be defined by

f(x)=e^{2x}

and let

g:(0,\infty)\to\mathbb{R}

be defined by

g(y)=\tfrac{1}{2}\ln y-3

. Denote by

h=g\circ f

. a. (i) Prove that

h: \mathbb{R}\to\mathbb{R}

is bijective. b. (ii) Find

h^{-1}

explicitly and state its domain and range. c. (iii) Determine, with justification, whether

f

is injective, surjective, or both relative to its declared domain and codomain. d. (iv) Determine, with justification, whether

g

is injective, surjective, or both relative to its declared domain and codomain.