Mathematics

Problem

Consider the two real-valued functions

f(x)=3^{x}

and

g(x)=\log_{3}x

. (i) Explain, without performing any numerical calculation, why

f

and

g

are inverse functions of each other. (ii) Using the fact established in part (i), justify why the domain of

g

equals the range of

f

and vice-versa. (iii) If the point

(p,\,q)

lies on the graph of

f

, explain algebraically which point must lie on the graph of

g

. (iv) On the same set of axes, sketch the qualitative shapes of

y=f(x)

and

y=g(x)

and clearly mark their common intercepts and asymptotes.