QUESTION
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, C1, and C2 such that C1|g(x)| ≤ |f (x)| ≤ C2|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 $x>k$.
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