Analytical Inverse function of exponential function of polynomials

Question

How can I obtain the inverse of the function below analytically?

$$e^{(0.0116t^2-0.4212t)},\ \ \ 0<t \leq 7.0633$$

Someone insists that it can be analytically obtained.

Answer 1

Let: $$y=e^{0.0116t^2-0.4212t}$$ Then, taking logarithms on both sides: $$\ln{y}=0.0116t^2-0.4212t$$ Therefore: $$0.0116t^2-0.4212t-\ln{y}=0$$ Note that you can now use the quadratic formula to solve explicitly for $t$.