Practice Problems on Function Inversion

For each of these functions f, find a candidate inverse function g (if possible!) and prove that the g=f^{-1}.

  1. f:\mathbb N\to \mathbb N. f(x)=2x+1

  2. f:\mathbb R\to \mathbb R. f(x)=x^2

  3. f:\mathbb N\to \mathbb N. f(x)=x^2

  4. f:\mathbb R\to \mathbb R. f(x)=x^3+x

  5. f:\mathbb R\to \mathbb R. f(x)=x^3-x

  6. f:\mathbb Z\to N,

    f(x)=\begin{cases} -2z & z\le 0\\ 2z-1 & z > 0 \end{cases}
  7. Find an example of a function function that is self-inverse (i.e. f=f^{-1}).

Worked sample problem:

f:\mathbb Z\to N,

f(x)=\begin{cases} -2z & z\le 0\\ 2z-1 & z > 0 \end{cases}

As discussed in class, this function maps the negative numbers to the even numbers and the positive numbers to the odd numbers. Here's a candidate inverse:

g:\mathbb Z\to N,

g(n)=\begin{cases} -\frac{n}2 & n\bmod 2=0\\ \frac{n+1}2 & n \bmod 2=1 \end{cases}

To show that g=f^{-1}, let z\in \mathbb Z and n\in \mathbb N. We'll then show f(z)=n\equiv z=g(n).

"\rightarrow" Let f(z)=n.

Thus g(n)=z in both cases.

"\leftarrow" Let z=g(n).

Thus f(z)=n in both cases.

This concludes the proof of everything required by the characterization of the inverse function.