Decide for which natural numbers the inequality3^n > n^3 is true.Prove your claim using mathematical induction Solution base case: n=4 3^(n+1)=3*3^n and (n+1)^3=n^3+3n^2+3n+1. What does 3^n+1/ (n+1)^3 equal? take the limit and use Lhopitals Rule toevaluate: lim, as n approaches infty, 3*3^n/(n^3 +3n^2+3n+1). Nowdivide the top and bottom by n^3. and you get the same limit aswith n. hope that helps.

1. Decide if it is a function from the domain to the codomaingiven? Decide whether is one-to-one
or onto? Descrive the inversefunction for any bijective function.
f: Z2 -->N where f is defined by f(x,y) =x2 + 2y2
Decide if it is a function from the domain to the codomaingiven? Decide whether is one-to-one
or onto? Descrive the inversefunction for any bijective function.
f: Z2 -->N where f is defined by f(x,y) =x2 + 2y2
Solution
QuestionDetails:
Decide if it is a function from the domain to the codomaingiven? Yes, it is a function from Z2
toN.
Decide whether is one-to-one or onto? Yes,it is 1-1. No, it is not onto. For example, there is
no(x,y) such that f(x,y)=5.
Descrive the inverse function for any bijective function.It is not onto, so it is notbijective.
f: Z2 -->N where f is defined by f(x,y) =x2 + 2y2