In my book, it is given that a function is one-to-one if its derivative is strictly monotonic. Otherwise, it is many-to-one. If this is the case, if we consider a function say f(x)= 2x + sin(x), the derivative is f '(x)= 2 + cos(x). The derivative is clearly non-monotonic, so the given function must be many-to-one. But when I used Desmos to plot the graph, I found it to be one-to-one and strictly increasing. How this is so. Please explain. Is the statement in my book incorrect?
Methods of determining whether a given function is one-to-one or many-to-one
functions
Best Answer
Let's stick to continuous, differentiable functions. A real function $f$ is one-to-one if it is strictly monotonic (the function itself, not its derivaitve!!)
You can guarantee that a function is strictly monotonic if its derivative does not change sign (let's ignore zeros for now. I mean: it either stays positive, or stays negative). In your case, $f'(x)=2+\cos{(x)}$ is always greater than zero.
It could be the case that $f'$ has a zero and $f$ is still strictly monotonic (think of $f(x)=x^3$), but what cannot happen is that there are points where $f'$ is positive and other points where $f'$ is negative