General Function question

1. Dec 10, 2013

AJKing

If a function s(t) exists, does a function t(s) always exist?

Are there functions with no inverse relationships?

Suppose

$s = \int^t_a e^{u^2} du$

Can there be a t(s)?

2. Dec 10, 2013

Staff: Mentor

Look at the trig functions for sin and cos over 0 to 2pi. They are clearly functions if inverted will map to two angles for a given sin or cos value. So the inverse is not a function.

For the sin if you restrict it 0 to pi/2 then its invertible and similarly for cos if you restrict it to 0 to pi.

http://en.wikipedia.org/wiki/Inverse_function

Try drawing a graph of e^u^2 and estimate the area under the curve and see if its invertible.

Last edited: Dec 10, 2013
3. Dec 10, 2013

Staff: Mentor

Let's revise your notation a bit to make things more understandable.
Suppose y = s(t) is a function. "s" is just the name of the function that maps values of t to values of y. Many functions do not have inverses that are themselves functions. A very simple example is y = f(x) = x2. Because f is not one-to-one, f does not have an inverse.

4. Dec 10, 2013

AJKing

Hmm, what about a solution involving step functions?

x = u0 y1/2-(1-u0)y1/2

Wolfram visual.

Last edited: Dec 10, 2013