Let where and is any real number

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

Problem

Let

f(x)=a x^{4}+b

where

a>0

and

b

is any real number, and let

g(x)=d x^{3}+e x

where

d\neq0

and

e

is any real number. Answer the following questions about the two general polynomials: (i) Show algebraically that

f(x)

is an even function and

g(x)

is an odd function. (ii) Describe the symmetry of the graphs of

y=f(x)

and

y=g(x)

that follows from part (i). (iii) Explain why

g(x)

must have at least one real zero, whereas

f(x)

may have none depending on the value of

b

. Base your explanation only on the symmetry and sign behaviour of the functions. (iv) Using degree and leading coefficient, compare the rate of growth of

|f(x)|

and

|g(x)|

|x|\to\infty

. State which function dominates and give a brief justification.

Related theory

Polynomial Functions
Functions

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