Problem: Determine whether the following function is 1-1, and if yes then find its inverse function. Also investigate monotonicity.
Solution: The domain of this function is the whole real line. To check
on injectivity we take the equation
In the second step we used the fact that arctan is a 1-1 function, so if the values are equal, the arguments must be equal as well. The same argument was used in the last step, this time using injectivity of the cube power. We see that the only solution is the trivial one, therefore this function is 1-1.
Now we find the inverse by solving
Remark:
When looking for the inverse function, we at one point had an equality
Note that this only works with arctangent. If we had
To investigate monotonicity by definition, we start with a couple
Note that in the second step we applied the cube power to both sides, which is a permissible operation, since x3 is an increasing function, that is, it turns smaller arguments into smaller values. For the same reason we were able to apply the increasing arctangent in the last step.
Anyway, we have shown that for any two numbers from the domain the function when applied preserves their order (smaller to smaller, larger to larger), so we in fact proved that this function is increasing on the whole real line.
Remark: For monotonicity we could also check on the sign of the derivative:
This is always positive and the conclusion follows, see Derivatives - Theory - MVT.
Note that once we have strict monotonicity on the domain, it follows that the function is 1-1 there.