Consider the functions

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

Problem

p(x)=2x+1

q(x)=x^{2}

and

r(x)=3^{x}

, defined for

x\ge 0

. (i) Without performing any numerical substitutions, explain which of the three functions grows the fastest as

x

becomes very large. (ii) Draw rough sketches of the graphs of

p

q

and

r

on the same set of axes for

0\le x\le5

. Label each graph clearly. (iii) Using your sketches, identify an interval of

x

within

0\le x\le5

where

q(x)

exceeds both

p(x)

and

r(x)

. Explain your observation.