Show that$$ f(x)=\frac{1}{2}\sin 2x + x$$is invertible.
How do I continue with this?
I've tried with taking the derivative and taken the fact that$$ f'(x)=\cos 2x +1 \ge 0.$$
Is this enough? I'm not sure since it can also equal $0$...
2026-04-02 16:56:21.1775148981
Show that the function $f(x)=\frac{1}{2}\sin 2x + x$ is invertible
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Depends how fussy you are. Suppose that $a\lt b$. We want to show that $f(a)\lt f(b)$.
Step a tiny amount to the right of $a$, say to $c$, where $c\lt b$ and there is no $x$ strictly between $a$ and $c$ such that $f'(x)=0$. Then $f(a)\lt f(c)$. Also, $f(c)\le f(b)$, so $f(a)\lt f(b)$.