Let be a non-zero real number and define two functions f: R R

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

Problem

Let

a

be a non-zero real number and define two functions

f: \mathbb R \to \mathbb R,\qquad f(x)=a\,x^{3},

g: \mathbb R \to \mathbb R,\qquad g(x)=\frac{\sqrt[3]{x}}{a}.

(a) Evaluate the composition

(f\circ g)(x)

and simplify the result. (b) Evaluate the composition

(g\circ f)(x)

and simplify the result. (c) Determine all real values of

a

for which both

(f\circ g)(x)

and

(g\circ f)(x)

are identity functions on their respective domains, thereby establishing that

f

and

g

are mutual inverses. (d) For any value of

a

that does not satisfy the condition found in part (c), provide a numerical example to illustrate that

f

and

g

fail to be mutual inverses.