Question

Show that \({x^3}\) is \(O({x^4})\) but that \({x^4}\)is not \(O({x^3})\).

Short answer

Hence, we obtain \({x^3}\) is \(O({x^4})\) but that \({x^4}\)is not \(O({x^3})\).

Step-by-step solution

Step 1:

Let\(f(x) = {x^3}\)and\(g(x) = {x^4}\)

Assuming\(k = 1\)

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

\(\begin{array}{l} = {x^3}\\ = 1.{x^3}\\ < x.{x^3}\\ = {x^4}\\ = \left| {{x^4}} \right|\end{array}\)

\( \Rightarrow C = 1\)

\( \Rightarrow f(x) = {x^3}\) is \(O({x^4})\)

Step 2:

Let\(f(x) = {x^4}\)and\(g(x) = {x^3}\)

Assuming\(f(x) = {x^4}\)is\(O({x^3})\)

\( \Rightarrow \)\(\left| {{x^4}} \right| \le C\left| {{x^3}} \right|\)when\(x > k\)

Let\(C \ge 0\)

Then,\(\left| {{x^4}} \right| \le C\left| {{x^3}} \right|\)\( = \left| {C{x^3}} \right|\)

Here, we are getting a contradiction because\(\left| {{x^4}} \right| > \left| {C{x^3}} \right|\)where\(x > k\)

\( \Rightarrow \)\({x^4}\)is \(O({x^3})\)