Question

Let $f\left(x\right)=\frac{a_{2k}{x}^{2k}+a_{2k-1}{x}^{2k-1}+....+a_{1}x+a_0}{b_{2k}{x}^{2k}+b_{2k-1}{x}^{2k-1}+....+b_{1}x+b_0}$
where $k$ is a positive interger, $a_i,b_i\in R$ and $a_{2k}\ne 0,b_{2k}\ne 0$ such that $b_{2k}{x}^{2k}+b_{2k-1}{x}^{2k-1}+.....+b_{1}x+b_0=0$ has no real roots, then

• A. $f\left(x\right)$ must be one to one
• B. $a_{2k}{x}^{2k}+a_{2k-1}{x}^{2k-1}+.....+a_{1}x+a_0=0$ must have real roots
• C. $f\left(x\right)$ must be many to one
• D. nothing can be said about the above options

Answer 1

The correct option is C $f\left(x\right)$ must be many to one

f(x) is continuous $\forall xϵR$.
$\underset{x\to -\infty }{lim}f\left(x\right)=\underset{x\to \infty }{lim}f\left(x\right)=\frac{a_{2k}}{b_{2k}}$
$\Rightarrow f\left(x\right)$ is many to one.
Hence, (c) is the correct answer.