Mathematics

Problem

For a non–zero real number

k

, consider the function

f:\mathbb R\to\mathbb R

defined by

f(x)=k x+(1-k).

A candidate function

g:\mathbb R\to\mathbb R

is given by

g(x)=\frac{x+k-1}{k}.

Answer the following questions. a. (i) Show that

f\bigl(g(x)\bigr)=x

for every real

x

, provided

k\ne 0

. b. (ii) Show that

g\bigl(f(x)\bigr)=x

for every real

x

, again assuming

k\ne 0

. c. (iii) Explain why the condition

k\ne 0

is necessary. Hence conclude for which real values of

k

the two functions are inverses of each other.