Consider the functions

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

f:\mathbb R\to\mathbb R

f(x)=ax+b

and

g:\mathbb R\to\mathbb R

g(x)=x^{2}+1

, where

a,b\in\mathbb R

. (i) Obtain explicit expressions for

(f\circ g)(x)

and

(g\circ f)(x)

. (ii) Determine all real values of

a

and

b

for which

(f\circ g)(x)=(g\circ f)(x)

for every

x\in\mathbb R

. Justify each step of your working.