Let be defined by and be defined by .

MATHEMATICS · Multi-part · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Let

f: \mathbb R \to \mathbb R

be defined by

f(x)=3x+2

and

g: \mathbb R \to \mathbb R

be defined by

g(x)=x^{3}

. (i) Find

(g\circ f)(x)

and simplify. (ii) Show that

(g\circ f)

is invertible and prove that

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

. Hence obtain an explicit formula for

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

. (iii) Verify that

(g\circ f)^{-1}\circ (g\circ f)=I_{\mathbb R}