I pass some colloquiums to find inverse functions. But still can't understand the real help of them. Only one real world example come to my mind: converting units of measurement (but those convertions should be injective).
Are there any other staff behind it when inverse functions helps a lot?
Best Answer
It may be helpful to note that finding inverse functions and solving equations is the same thing, if you look at it from the right perspective.
For example, take the simplest physical system you can think of, say an object being thrown up with velocity $v$, and then falling freely. (Maybe that's not strictly speaking the simplest one, but bear with me.) Then the height at which the object is at time $t$, say $h(t)$, is given by $h(t) = vt - \frac{1}{2}gt^2$, if I remember my highschool physics correctly. Suppose you want to figure out at which time this object reaches a given height $h_0$. One way of looking at it is solving the equation $vt - \frac{1}{2}gt^2 = h_0$ and finding a time $t_0$ at which this happens. But you may just as well look at the inverse function $h^{-1}$, and then take $t_0 = h^{-1}(h_0)$.
On a slightly more advanced level, there is a theorem known as the Inverse function theorem, which tells you basically that a function $F$ can be (locally) inverted, subject to some natural restrictions (and that the inverse is 'nice' if the function is 'nice'). There is also a Implicit function theorem which tells you that if you have an equation $R(x,y) = 0$ ($x,y$ standing for a bunch of variables, not just one, and $R$ taking values in $\mathbb{R}^d$, so we are really looking at $d$ equations), then you can 'solve for $y$' and find $y$ given by $y = f(x)$ (where $f$ is a 'nice' function, provided that the equations were 'nice'). I hope you will grant me that the inverse function is useful in 'real life'.
Now, these theorems are very closely related. If you know about the Inverse function theorem, you can prove the Implicit function theorem just by looking at $F(x,y) = (x,R(x,y))$; then $y$ is given by $f(y) = \pi_2(F^{-1}(x,0))$, where $\pi_2(x,y) = y$.