Let be defined by .

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Let

p: \mathbb R \to \mathbb Z

be defined by

p(x)=\Bigl\lfloor \log_{2}\bigl(x^{2}+1\bigr)\Bigr\rfloor

. (i) Express

p

as a composition of two named standard functions and state clearly which two they are. (ii) Determine all real numbers

x

for which

p(x)=0

. (iii) Determine all real numbers

x

for which

p(x)=1

p(x)=2

, and

p(x)=3

respectively. (iv) Sketch the graph of

p

for

0\le x\le 3

, carefully indicating the value taken on each interval. (v) State the complete domain and range of

p