For a non–zero real number

MATHEMATICS · Subjective · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

a

, define

f_a:\mathbb R\to\mathbb R

f_a(x)=ax+1

, and let

g: \mathbb R\setminus\{0\}\to\mathbb R\setminus\{0\}

be given by

g(x)=\dfrac{1}{x}

. (a) Determine all real values of

a

for which the composition

g\circ f_a

is invertible as a function from its natural domain to its codomain. (b) For each admissible value of

a

, obtain an explicit formula for

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

. (c) Taking

a=2

, verify that

(g\circ f_2)^{-1}\bigl(g\circ f_2(x)\bigr)=x

for every

x

in the domain of

g\circ f_2