For the logarithmic function a new function is defined by g(x)=-3 , _ 10 (5-x)+1

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

For the logarithmic function

f(x)=\log_{10}(x-2)

a new function is defined by

g(x)=-3\,\log_{10}(5-x)+1.

Answer the following questions related to the two functions. (i) Describe, with reasons, the horizontal transformation that maps the argument of

f

to the argument of

g

. Hence explain how this transformation affects the domain. (ii) Determine the exact domain of

g

. (iii) The range of

f

\mathbb R

. Explain how the vertical stretch, reflection and translation used in defining

g

influence its range, and state the range of

g