The given function is:

$f(x) = \frac{1}{x - 2} + 2x^2 - 2x^2$

Upon simplifying, note that $2x^2 - 2x^2 = 0$, so the function reduces to:

$f(x) = \frac{1}{x - 2}$

Finding Roots

To find the roots of $f(x)$, set $f(x) = 0$:

$\frac{1}{x - 2} = 0$

There is no solution for this equation because $\frac{1}{x - 2}$ can never be zero for any value of $x$. Thus, there are no roots for this function.

Finding Vertical Asymptote

The vertical asymptote occurs where the denominator of the function is zero. Set $x - 2 = 0$:

$x = 2$

So, there is a vertical asymptote at $x = 2$.

Summary

• No roots.
• Vertical asymptote at $x = 2$.

Would you like more details or have any other questions?

Related Questions

1. How do we find horizontal asymptotes for rational functions?
2. What is the general procedure to find vertical asymptotes in any function?
3. Can a function have more than one vertical asymptote?
4. What is the significance of a root in the context of a rational function?
5. How would adding other polynomial terms change the asymptotic behavior?

Tip

Always analyze the simplified form of a function to clearly identify roots and asymptotes.