Function to change inputs from 0.5 to 1, to 0 to 1

Question

Let's say we have x, which is any decimal number between 0.5 and 1.

What function can be applied to it such that an input of 0.5 equals 0 and input of 1 equals 1? It should be linear.

I tried something like this:

f(x) => x / 2 + 0.5

Which works for 1, but not 0.5. Apparently my math skills are lacking... What is the approach to figure out problems like this in general?

PS - for the curious, this is the value of the opacity of a label while paging items in an iPad app.

Answer 1

Suppose the function we want to find is $f(x)$. Since it should be linear, we have $f(x)=ax+b$. Suppose we want $f(x_0)=y_0$ and $f(x_1)=y_1$. That can be rewritten to $$ ax_0+b=y_0\\ ax_1+b=y_1 $$ Solving that for $a$ and $b$ yields $$ a=\frac{y_0-y_1}{x_0-x_1}\\ b=y_0-ax_0 $$ In your case, we have $f(\frac 12)=0$ and $f(1)=1$. That gives $$ a=\frac{1-0}{\frac 12-0}=\frac 1{\frac 12}=2\\ b=0-2\frac 12=-1\\ f(x)=ax+b=2x-1 $$

Answer 2

First look at $h(x)=x-\frac{1}{2}$. That will make the input go from $0$ to $\frac{1}{2}$. So what you want is actually

$$g(x)=2\cdot h(x)=2x-1.$$