**QUESTION**

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Identify the correct steps involved in the proof of the statement "If *f*(*x*) and *g*(*x*) are functions from the set of real numbers to the set of real numbers, then *f*(*x*) is Θ(*g*(*x*)) if and only if there are positive constants* k*, *C*_{1}, and *C*_{2} such that *C*_{1}|*g*(*x*)| ≤ |*f* (*x*)| ≤ *C*_{2}|*g*(*x*)| whenever *x* > *k*." (Check all that apply.)

Check All That Apply Suppose $f(x)$ is $\Theta(g(x))$. Then, $f(x)$ is both $Q(g(x))$ and $\Omega(g(x))$. There exists positive constants $C_{1}, k_{1}, C_{2}$, and $k_{2}$ such that $|f(x)| \leq C_{2} \mid g\left(x||\right.$ for all $x>k_{2}$ and $|f(x)| \geq C_{1}|g(x)|$ for all $x>k_{1}$. Let $k=\max \left\{k_{1}, k_{2}\right\}$. Then, for $x>k, C_{1}|g(x)| \leq|f(x)| \leq C_{2}|g(x)|$. Let $k=\min \left\{k_{1}, k_{2}\right\}$. Then, for $x>k, C_{1}|g(x)| \leq|f(x)| \leq C_{2}|g(x)|$. Suppose there exists positive constants $C_{1}, C_{2}$, and $k$ such that $C_{1}|g(x)| \leq|f(x)| \leq C_{2}|g(x)|$ whenever $x>k$. Let $k=k_{1}=k_{2}$. Then, $|f(x)| \leq C_{2}|g(x)|$ for all $x>k_{2}$ and $|f(x)| \geq C_{1}|g(x)|$ for all $x>k_{1}$. Let $k=k_{1}=k_{2}$. Then, $|f(x)| \leq C_{2}|g(x)|$ for all $x>k$ and $|f(x)| \leq C_{1} \mid g(x||$ for all $x>k$.