$f(x)=x^3-x$
The question was the prove $f(x)$ is not injective and is surjective. I didn't manage to prove the latter. Is that a particular technique to prove $y=f(x)$ to show it's surjective or something?
Thanks.
$f(x)=x^3-x$
The question was the prove $f(x)$ is not injective and is surjective. I didn't manage to prove the latter. Is that a particular technique to prove $y=f(x)$ to show it's surjective or something?
Thanks.
One way is to use the intermediate value theorem.
First, prove that $f(x)$ is continuous.
Then show for any $y$ you can make find $x_1$ so that $f(x_1)>y$ and $x_2$ so that $f(x_2)
Not injective since $x = -1$, $x = 0$, and $x = 1$ all map to $y = 0$. It is sufficient to find two points in the domain that map to the same point in the codomain.
For surjective, this could easily be seen by plotting, but I agree with the previous response.