Two functions are defined for all real numbers: and .

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

Problem

Two functions are defined for all real numbers:

p(x)=3x^{5}+7x^{3}-x

and

q(x)=2^{x}

. (i) Identify and name the parent family to which each function belongs. (ii) Show that

\displaystyle \lim_{x\to\infty}\frac{q(x)}{p(x)}=\infty

. (iii) Explain how the result in part (ii) validates the classifications made in part (i). (iv) State whether

q(x)

eventually becomes less than

p(x)

for sufficiently large negative values of

x

, supporting your answer with brief reasoning.