Let
f ( x )

To analyze the behavior of the function $f(x) = 2x + 4x^2 - 2x$ as $x$ gets larger, we can examine the leading term, which is the term with the highest degree in $x$. This approach helps us understand how $f(x)$ behaves as $x \to \infty$.

Step 1: Identify the Leading Term

The function $f(x)$ can be simplified to: $f(x) = 4x^2$ since the linear terms $2x$ and $-2x$ cancel each other out.

Thus, the leading term is $4x^2$, which dominates the function’s behavior for large values of $x$.

Step 2: Analyze the Leading Term as $x \to \infty$

As $x$ grows larger, $4x^2$ will also grow larger in a positive direction, since $4x^2$ is positive for all values of $x$. Therefore, as $x \to \infty$, the values of $f(x)$ will also tend toward infinity.

Conclusion

As $x$ gets larger, the function values $f(x)$ will increase without bound, getting closer to $+\infty$.

Would you like me to verify this result using a table of values or a graph for further confirmation?

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Here are 5 related questions to expand your understanding:

1. How would the function behave as $x \to -\infty$?
2. What is the effect of adding a constant to $f(x)$ on the end behavior of the function?
3. How do you determine which terms in a polynomial dominate as $x \to \infty$?
4. If the coefficient of $x^2$ was negative, how would the function behave as $x \to \infty$?
5. What role do non-leading terms play in determining the function’s behavior for small values of $x$?

Tip: For large values of $x$, focus on the highest degree term in a polynomial, as it usually dictates the function's end behavior.