For a real parameter

MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

k\,(k\neq0)

, consider the family of functions

f_{k}(x)=k\,2^{x}

. (i) Prove that the range of

f_{k}

(0,\infty)

when

k>0

and

(-\infty,0)

when

k<0

. (ii) Hence determine all real values of

k

for which the range of

f_{k}

contains the number

3

. (iii) If

k=-6

, a vertical translation of

f_{k}

8

units upwards gives

s(x)=-6\,2^{x}+8

. Find the domain and range of

s(x)

, justifying every statement.