# Inverse function question

Greetings all. I was solicited by a friend to find the inverse of a particular function, and I can't for the life of me determine/remember how.

The original equation is
y = 3+x^2+tan((1/2)*Pi*x)
with x on (-1,1).

The function is invertible - f' is always > 0 on that interval - but I have had no success attempting to determine precisely what the inverse is.

The graph of the inverse function is easy to generate: reflect the graph of the existing function over the line y = x. You can also easily find the derivative of the inverse function and thus a Taylor series for the inverse function.
However, it is not necessarily possible to write the inverse function of a function in terms of a finite combination of elementary functions, especially when transcendental functions are involved (ie., trigonometric, exponential and logarithm functions).

The graph of the inverse function is easy to generate: reflect the graph of the existing function over the line y = x. You can also easily find the derivative of the inverse function and thus a Taylor series for the inverse function.
However, it is not necessarily possible to write the inverse function of a function in terms of a finite combination of elementary functions, especially when transcendental functions are involved (ie., trigonometric, exponential and logarithm functions).

I thought this as well - obtaining the graph of the inverse function would seem to be more within the scope of a first-week Pre-Calculus course - but it seems that the function itself is what is required.

I am sorry to say that short of some serious Taylor series wrangling, the function you are looking for is not elementary and cannot be found in closed form.

--Elucidus