Some gaps in our reasoning

First of all, with the exception of circles, the slope of tangent lines were defined to be the rates of change, so we cannot ``go backwards''. Second of all, even in our algebraic derivation for roots we needed that roots were continuous, that is that limits of roots can be found by evaluation. This we will show to be true later. The more general theorem, called The Inverse Function Theorem, which we will not prove in this course, says

Inverse Function Theorem: Suppose that $f:[a,b]\rightarrow (-\infty,\infty)$ is continuous and invertible. Then the range of f is also a closed interval, call it [c,d]. $f^{-1}:[c,d]\rightarrow [a,b]$ is also continuous. In addition, f^-1 will be differentiable at a if

1.

f is differentiable at f^-1(a);

2.

$f'(f^{-1}(a)) \neq 0$;

and

$$(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}.$$

You should observe that the only instance in which we have actually verified the hypotheses of this theorem is in the case of functions with polynomial rules. In a course called Advanced Calculus a proof of this theorem will be given, as well as proofs that functions like sine, cosine, and the like, satisfy its hypotheses.