Linear to Nonlinear Function with Minimum and Maximum

I am currently in Calc 1 and I am not qute sure if this is even Calc and/or how it can be solved. Please let me know if I am not clear at any point in this.

The type of application for this is:

JS Bin

I would use that one, but it doesn't work on anything but that particular example from 1-100, I need it to work from A to B as shown below.

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So I have a slider (see above) with the function:

f(x) = x

So when you side the slider it will be a linear increase/decrease with a variable minimum and maximum for this slider, so when it is at the far left it is at point A and at the far right, point be so we can say:

A < B

I need a function (call it g(x)) that does the following:

f(A) = g(A)

and

f(B) = g(b)

but here is the twist

Note a variable D where:

A < D < B

f(D) > g(D)

So what I want is for f(x)=x to have a g(x)=x^2 type action to be scaled to fit the minimum and maximum.

Let me know if I am not being clear.

Thanks in advance!