I have been tasked with checking if the following functions are injective, surjective or bijective:
(a) $f:\mathbb {N} \to \mathbb {N} ,n \to n^4$
(b) $f:\mathbb {R} \to [−1,1], x \to \sin(x)$
Now, I have no idea how to approach these functions since there are no numbers included. I know that for a function to be injective, for $f(x_1)=f(x_2)$ has to apply $x_1=x_2$. However, I have no idea what to plug here since there is only $n^4$.
My guess for (a) is that it is injective, while on (b) I am completely stumped.
Best Answer
For injectivity in part (a), if $f(x_1)=f(x_2)$, then by the definition of $f$, we have $x_1^4=x_2^4$. Does this imply that $x_1=x_2$ (keeping in mind that $x_1$ and $x_2$ are nonnegative integers)? For surjectivity, is it true that any $m \in \mathbb{N}$ satisfies $m=n^4$ for some $n \in \mathbb{N}$?
For injectivity in part (b), if $f(x_1)=f(x_2)$, then $\sin(x_1)=\sin(x_2)$. Does this imply that $x_1=x_2$ (where now $x_1$ and $x_2$ can be any real numbers)? For surjectivity, is it true that for any $y \in [-1,1]$ there is $x \in \mathbb{R}$ such that $\sin(x)=y$?