The function is given by

MATHEMATICS · Subjective · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

The function

p

is given by

p(x)=\log_{b}(kx+c)

, where

b>0

b\neq1

k\neq0

, and

c

are real constants. (a) Derive an explicit expression for

p^{-1}(x)

in terms of

b

k

, and

c

. (b) List the conditions that

k

and

c

must satisfy so that both

p

and

p^{-1}

are defined for real inputs. (c) Prove algebraically that

p\bigl(p^{-1}(x)\bigr)=x

for all

x

in the domain of

p^{-1}

. (d) Taking

b=\tfrac12

k=3

, and

c=-6

, state the domain and range of the corresponding inverse function.