Mathematics

Problem

Let

a

be a real number satisfying

a>0

and

a\ne 1

. Two functions are defined as follows:

f:\mathbb R\to(0,\infty),\qquad f(x)=a^x,

g:(0,\infty)\to\mathbb R,\qquad g(x)=\log_a x.

Answer the following questions. a. (i) Verify that

f\bigl(g(x)\bigr)=x

for every

x>0

. b. (ii) Verify that

g\bigl(f(x)\bigr)=x

for every real

x

. c. (iii) State the domain and range of each function and explain briefly how these sets are interchanged between

f

and

g

. Hence confirm that

f

and

g

are inverses of each other.