Cantor's bijection between the sets [0,5] and [12,60]

Question

How can I mathematically prove that the sets $[0,5]$ and $[12,60]$ have the same cardinality using Cantor's bijection? While simply drawing a linear function seems tempting, my teacher wants us to do so by demonstrating the four following statements:

1. The function is injective.
2. The function is surjective.
3. If $x \in [0,5]$, then $y\in [12, 60]$.
4. If $y \in [12,60]$, then $x \in [0,5]$.

Answer 1

Hint: Linear interpolation polynomial.

Some details:

The linear polynomial that takes the value $12$ at $x=0$ and $60$ at $x=5$ is: $$l(x)=60\frac{x-0}{5-0}+12\frac{x-5}{0-5}.$$