The task is to determine whether the function $3x^2 + x + 1$ is $\Theta(x^2)$, which means proving that there exist positive constants $k$, $C_1$, and $C_2$ such that:

$C_1 g(x) \leq f(x) \leq C_2 g(x)$

for all $x > k$, where $g(x) = x^2$ and $f(x) = 3x^2 + x + 1$. To do this, the steps should establish the lower and upper bounds on the function.

Let me guide you through the steps in order:

1. Step 1: Take $k = 1$, $C_1 = 1$, and $C_2 = 2$.

   • This is the setup of constants to satisfy the Big-O definition.

2. Step 2: $3x^2 \leq 3x^2 + x + 1 \leq 3x^2 + 3x^2 = 6x^2 \text{ for all } x > 1$.

   • Establishes that the given function $f(x)$ is bounded by $x^2$ terms on both sides.

3. Step 3: $3x^2 \leq 3x^2 + x + 1$ for $x > 0$.

   • Demonstrates that $f(x)$ grows at least as fast as $3x^2$, showing the lower bound.

4. Step 4: $3x^2 + x + 1 \leq 6x^2 \text{ for all } x > 1$.

   • Shows the upper bound by comparing $f(x)$ with a constant multiple of $x^2$.

5. Step 5: Thus, $3x^2 + x + 1$ is $\Theta(x^2)$.

   • Concludes that the function satisfies both the upper and lower bounds, meaning it is indeed $\Theta(x^2)$.

Does this explanation make sense to you? Would you like further details?

Relative Questions:

1. What does it mean for a function to be $\Theta(x^2)$?
2. How do we find constants for Big-O and Big-$\Omega$ proofs?
3. Why is it necessary to prove both upper and lower bounds in asymptotic notation?
4. Can a function be both $O(x^2)$ and $\Omega(x^2)$ without being $\Theta(x^2)$?
5. What are some examples of functions that are not $\Theta(x^2)$?

Tip:

Always check both upper and lower bounds when working with $\Theta$-notation, as it provides a tight bound on the function’s growth rate.