Let be bijective functions such that the composite function h(x)=(f g)(x)= 2x-3 x+4

MATHEMATICS · Subjective · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Let

f, g: \mathbb R \to \mathbb R

be bijective functions such that the composite function

h(x)=(f\circ g)(x)=\frac{2x-3}{x+4},\qquad x\neq -4.

Further, the inverse of

g

is known to be

g^{-1}(x)=\frac{5x+2}{1-x},\qquad x\neq1.

(a) Show that

f(x)=h\bigl(g^{-1}(x)\bigr)

and hence obtain an explicit formula for

f(x)

, stating any restrictions on

x

. (b) Using your result from Part (a), determine

f^{-1}(x)

and its domain. (c) Verify directly that

f^{-1}\!\circ f

and

f\circ f^{-1}

are identity functions on their respective domains.