Construct a non-linear function that shows that the intervals $[2,4]$ and $[10,22]$ have the same cardinality

Question

Using something other than a linear function, show the intervals $[2,4]$ and $[10,22]$ have the same cardinality.

I don't quite know where to start with this problem, or what key factor is necessary to perceive the function that will satisfy the given problem.

Answer 1

Hint: You know $x^2$ is a bijection from $[0,1]$ to $[0,1]$. Translate and scale to produce the bijection you want.

$12x^2$ a bijection from $[0,1]$ to $[0,12]$.
$10 + 12x^2$ a bijection from $[0,1]$ to $[10,22]$.
Scale to get $10 + 12\left(\frac{x}{2}\right)^2$ a bijection from $[0,2]$ to $[10,22]$.
Translate to get $10 + 12\left(\frac{x-2}{2}\right)^2$ a bijection from $[2,4]$ to $[10,22]$

$3(x-2)^2 + 10$ is an answer.

Answer 2

Here is a recipe to turn a linear bijection between (almost) any two sets (for which being linear makes sufficient sense) into a non-linear bijection between the same two sets. Suppose $f\colon A\to B$ is a linear bijection. Pick two distinct points $p_1,p_2$ in $A$ (so we are assuming $|A|\ge 2$). The function $g\colon A\to B$ given by $g(p_1)=f(p_2)$, $g(p_2)=f(p_1)$, and $g(x)=f(x)$ otherwise is obviously bijective, but not linear if $|A|\ge 4$.

Answer 3

Here is a piece-wise linear map:

$f(x)= 10(x-1)\quad x\in [2,3]$

$f(x) = 2(x-2)+18\quad x\in[3,4]$