Mathematics

Problem

Let

f:\mathbb R\to\mathbb R

be defined by

f(x)=2x-3

and let

g:\mathbb R\to\mathbb R

be defined by

g(x)=x^{3}

. (i) Find

(g\circ f)(x)

. (ii) Determine

(g\circ f)^{-1}(x)

. (iii) Verify your result by showing that

\bigl((g\circ f)^{-1}\circ(g\circ f)\bigr)(x)=x

for all

x

in the domain of

f