a) Suppose $n$ is even. Are the following statements true or false for every polynomial of degree $n$? If true, select "NONE" from the list of possible counterexamples. If false, select all functions that are counterexamples to the statement.

i) $f(x)$ is an invertible function.
? True False

A. $x^2 + 4x + 3$
B. $-x^4 -5x +3$
C. $-x^3$
D. $x^5 -5x +3$
E. $-x +3$
F. NONE

ii) If $\displaystyle \lim_{x\to -\infty} f(x) = -\infty$, then $\displaystyle \lim_{x\to \infty} f(x)=-\infty$.

? True False

A. $x^2 + 4x + 3$
B. $-x^4 -5x +3$
C. $-x^3$
D. $x^5 -5x +3$
E. $-x +3$
F. NONE

iii) $\displaystyle \lim_{x\to \infty} f(x) = -\infty$

? True False

A. $x^2 + 4x + 3$
B. $-x^4 -5x +3$
C. $-x^3$
D. $x^5 -5x +3$
E. $-x +3$
F. NONE

b) Suppose $n$ is odd. Are the following statements true or false for every polynomial of degree $n$? If true, select "NONE" from the list of possible counterexamples. If false, select all functions that are counterexamples to the statement.

i) $f(x)$ is an invertible function.
? True False

A. $x^2 + 4x + 3$
B. $-x^4 -5x +3$
C. $-x^3$
D. $x^5 -5x +3$
E. $-x +3$
F. NONE

ii) If $\displaystyle \lim_{x\to \infty} f(x) = \infty$, then $\displaystyle \lim_{x\to -\infty} f(x)=-\infty$.

? True False

A. $x^2 + 4x + 3$
B. $-x^4 -5x +3$
C. $-x^3$
D. $x^5 -5x +3$
E. $-x +3$
F. NONE

iii) $\displaystyle \lim_{x\to \infty} f(x) = \infty$

? True False

A. $x^2 + 4x + 3$
B. $-x^4 -5x +3$
C. $-x^3$
D. $x^5 -5x +3$
E. $-x +3$
F. NONE

Hint: