Define three bijective functions by f(x)=3x-4

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

Problem

Define three bijective functions by

f(x)=3x-4,\qquad g(x)=\frac{1}{x-2},\quad x\neq2,\qquad h(x)=\sqrt{x},\quad x\ge0.

(i) Determine the domains and ranges of

f, g,

and

h

. (ii) Find the composite function

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

and state its domain. (iii) Obtain an explicit expression for

k^{-1}(x)

, clearly indicating the domain on which it is defined. (iv) Confirm that

k^{-1}(x)

can be written as

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

by evaluating the right-hand side and showing it agrees with your answer in Part (iii).