Complete the proof that any two open intervals $(a, b)$ and $(c, d)$ are equivalent
by showing that $f(x) = \frac{d-c}{b-a}(x-a) + c$ maps one to one and onto $(c,d)$. I showed one to one by saying that there exists $x$ that in $(a,b)$ and $y$ in $(c,d)$ st $f(y) = f(x )$ iff $x = y$. But I'm not sure how to show that it is onto, I thought I had to make $x$ in terms of y and then plug it into $f(x)$ but if that is the correct way of doing it, then I'm stuck on the algebra.
[Math] Showing that two intervals are equivalent.
elementary-set-theoryproof-verification
Best Answer
Just retrace the steps that the function is taking, and isolate $x$.
Pick $y\in(c,d)$. Consider $y-c$ first, then $(y-c)\frac{b-a}{d-c}$, finish by isolating $x$. Then show that $f(x)=y$.