https://file.aiquickdraw.com/mathbot/file/1728097021179fgfz7duo.jpg

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.

Find roots and vertical asymptote for the function f(x) = 1/(x - 2) + 2x^2 - 2x^2.

Learn how to find the roots and vertical asymptote for the function f(x) = 1/(x - 2). This problem involves simplifying the function and analyzing rational functions. Suitable for students in Grades 10-12.

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