Normalizing from [0.5 - 1] to [0 - 1]

Question

I'm kind of stuck here, I guess it's a bit of a brain teaser. If I have numbers in the range between 0.5 to 1 how can I normalize it to be between 0 to 1?

Thanks for any help, maybe I'm just a bit slow since I've been working for the past 24 hours straight O_O

Answer 1

Others have provided you the formula, but not the work. Here's how you approach a problem like this. You might find this far more valuable than just knowning the answer.

To map [0.5, 1] to [0, 1] we will seek a linear map of the form x -> ax + b. We will require that endpoints are mapped to endpoints and that order is preserved.

Method one: The requirement that endpoints are mapped to endpoints and that order is preserved implies that 0.5 is mapped to 0 and 1 is mapped to 1

a * (0.5) + b = 0 (1)
a * 1 + b = 1     (2)

This is a simultaneous system of linear equations and can be solved by multiplying equation (1) by -2 and adding equation (1) to equation (2). Upon doing this we obtain b = -1 and substituting this back into equation (2) we obtain that a = 2. Thus the map x -> 2x - 1 will do the trick.

Method two: The slope of a line passing through two points (x1, y1) and (x2, y2) is

(y2 - y1) / (x2 - x1).

Here we will use the points (0.5, 0) and (1, 1) to meet the requirement that endpoints are mapped to endpoints and that the map is order-preserving. Therefore the slope is

m = (1 - 0) / (1 - 0.5) = 1 / 0.5 = 2.

We have that (1, 1) is a point on the line and therefore by the point-slope form of an equation of a line we have that

y - 1 = 2 * (x - 1) = 2x - 2

so that

y = 2x - 1.

Once again we see that x -> 2x - 1 is a map that will do the trick.