How to map intervals of Real line

[Diffy]

Can someone explain how to create a function that will map an interval of the real line onto some other interval?

Is there a general method?

Can you demonstrate? (30 140) to (200, 260)?

Thanks,

Diffy.

 

[arildno]

try with a linear map.

 

[Diffy]

Let x1 be in (30, 140) and let x2 be in (200, 260)
then f: x1 -> x2
f(x) = (6/11)(x-30) + 200

Something like that?

Will this always work for (x1, x2) -> (y1, y2)?

f(x) = (y2 - y1)/(x2 - x1) * (x - x1) + y1

 

[arildno]

Indeed, let us prove it as follws:

1. For any X so that x1<=X<=x2, we have y1<=f(X)<=y2
Proof:
We clearly have :
y1=(y2-y1)/(x2-x1)*(x1-x1)+y1<=f(X)<=(y2-y1)/(x2-x1)*(x2-x1)+y1=y2

2. Similarly, we can show that for any Y so that y1<=Y<=y2, then there exists some X, so that x1<=X<=x2 AND that f(X)=Y.

We get that X=(Y-y1)/(y2-y1)*(x2-x1)+x1. Since 0<(Y-y1)/(y2-y1)<1, that result follows immediately.

 

[Diffy]

Thank you.

 

[Ben Niehoff]

In most cases, you can use a linear map. Some trickiness results if one of the intervals is infinite or half-infinite. In that case, you might have to use exp(), tan(), or their inverses.