Question

Show that \(3{x^4} + 1\) is \(O({x^4}/2)\) and \({x^4}/2\)is not \(O(3{x^4} + 1)\).

Short answer

Hence, we obtain\(3{x^4} + 1\)is\(O({x^4}/2)\)and\({x^4}/2\)is not\(O(3{x^4} + 1)\).

Step-by-step solution

Step 1:

Let\(f(x) = 3{x^4} + 1\)and\(g(x) = \frac{{{x^4}}}{2}\)

Assuming\(k = 1\)

Then,\(\left| {f(x)} \right| = \left| {3{x^4} + 1} \right|\)

\( \le \left| {3{x^4}} \right| + \left| 1 \right|\)

\( \Rightarrow C = 8\)

\( \Rightarrow f(x) = 3{x^4} + 1\) is \(O(\frac{{{x^4}}}{2})\)

Step 2:

Let\(f(x) = \frac{{{x^4}}}{2}\)and\(g(x) = 3{x^4} + 1\)

Let\(k = 1\)

\(\begin{array}{l} \Rightarrow \left| {f(x)} \right| = \left| {\frac{{{x^4}}}{2}} \right|\\ = \frac{{{x^4}}}{2}\end{array}\)

\( \Rightarrow \)\(f(x) = \frac{{{x^4}}}{2}\)is \(O(3{x^4} + 1)\)