Mathematics

Problem

Consider the functions

f: \mathbb R\setminus\left\{\tfrac12\right\} \to \mathbb R\setminus\left\{\tfrac32\right\},\qquad f(x)=\frac{3x+5}{2x-1},

and

g: \mathbb R\setminus\left\{\tfrac32\right\} \to \mathbb R\setminus\left\{\tfrac12\right\},\qquad g(x)=\frac{x+5}{2x-3}.

(a) Determine the range of

f

and verify that it equals the domain of

g

. (b) Show algebraically that

(g\circ f)(x)=x

for every admissible

x

. (c) Show algebraically that

(f\circ g)(x)=x

for every admissible

x

. (d) Hence establish that

f

and

g

are inverse functions of each other, clearly stating the domains and ranges for which the conclusion holds.