Is it true that **x^3** is **O(g(x))**, if **g** is the given
function?
- x^2
- x^3
- x^2+x^3
- x^2+x^4
- 3^x
- x^3/2

It should, by now, be clear that the answer is **yes** for all but
possibly the first
(in each case **C=2** and **k=3** from
the definition will work). Note, for **g(x)=3^x** we could say the
following: we can take for granted that **x^3** is **O(2^x)**
and **2^x< 3^x** for all **x> 1**, hence we are done by
Exercise 17 (which you should now do).
Let us check that the answer is
**no** for **g(x)=x^2**
Similar to previous problems, from the definition, can there be some
**C,k** such that for all **x> k**

** x^3< C*x^2**? Divide both sides by **x^2** (which is OK so
long as **x** is bigger than 0) we get

** x< C** which obviously does not hold for all **x> k**.