Answer the following with reference to polynomial functions of the form

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

Problem

f(x)=a x^{n}

, where

a\neq 0

and

n\in\mathbb{N}

. (i) State the type of symmetry exhibited by the graph of

f(x)

when

n

is even and when

n

is odd. (ii) For

a>0

, describe the behaviour of

f(x)

x\to+\infty

and as

x\to-\infty

separately for the cases

n

even and

n

odd. (iii) Repeat part (ii) for

a<0

, clearly indicating the limit-values and the qualitative position of the graph in each direction.