Finding if a function is one-one and onto Consider the function $f:\mathbb R \to \mathbb R$ given by $f(x)= |x| (x - \sin x)$. How do I prove that this function is one-one. Is there any method apart from checking if the function is increasing / decreasing?
Study the function for when $x \ge 0$ and for when $x&lt;0$. You immediately see your function is odd (so it’s symmetric with respect to the origin) so you can study its injectivity only for the $x \ge 0$ part. Then suppose $f(x)=f(y)$ for some $x,y$ and try to conclude that your equation has solutions only if $x=y$.
Beside the fact that here the monotonicity test is way easier.
Study the function for when $x \ge 0$ and for when $x&lt;0$. You immediately see your function is odd (so it’s symmetric with respect to the origin) so you can study its injectivity only for the $x \ge 0$ part. Then suppose $f(x)=f(y)$ for some $x,y$ and try to conclude that your equation has solutions only if $x=y$.
Beside the fact that here the monotonicity test is way easier.