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 f (x1) = f (x2) and see whether there is other than the trivial solution.

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 y = f (x) for x.

Remark: When looking for the inverse function, we at one point had an equality y = arctan(z) and changed it into z = tan( y).

Note that this only works with arctangent. If we had y = tan(z), then it is not possible to conclude that z = arctan( y)! The reason is that to find the inverse to tangent we had to restrict the domain of tangent to . The tangent and arctangent are mutual inverses only assuming that the variable in tangent is from this interval. In the equation y = arctan(z) is follows automatically that y is from this interval, since it is the value of arctan. On the other hand, given the equation y = tan(z), there is no guarantee that z is from that interval and so the inverse relationship between tangent and arctangent does not apply.

To investigate monotonicity by definition, we start with a couple x1 < x2 from the domain and see whether we can establish any inequality between function values. This we try by manipulating the inequality to create the given function on both sides using elementary operatios.

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.


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