How do I show a function is Big-O of another function using the definition of Big-O? **Definition** : A function F(x) is Big-O of g(x) if we can find constant witnesses such that $f(x) <= Cg(x)$ when $x=k$. Use the definition of “$f (x)$ is $O(g(x))$” to show that: $x^4 + 9x^3 + 4x + 7$ is $O(x^4)$ I tried dividing both sides by $x^4$, but I'm not sure how to find a tight bound without repeatedly guessing and checking. Another way to phrase this problem: After modifying C, so that $g(x)$ approximates $f(x)$, how would one find the intersection of the two functions. P.S. I'm studying for a test so I'm looking for how to solve these problems in general.

The definition is that exists some constant $C>0$ such that $$\left|f\left(x\right)\right|\leq Cg\left(x\right)$$ as $x\rightarrow x_{0}$ , where $x_{0}$ can be $\infty.$ So I think you're interessed when $x\rightarrow\infty.$ In this case it's sufficient to note that $x^{4}$ grow up faster then other power of $x$, so $$x^{4}+9x^{3}+4x+7\leq x^{4}\left(1+9+4+7\right)=21x^{4}.$$ Note that if $x_{0}=0$, for example, this argument doesn't work, so be careful about $x_{0}.$