text: user14989: When we have a function which takes values ​​in [a,b] and which returns values ​​in [c,d], what transformation can we do so that the function takes values ​​in [0,1] and returns values ​​in [0,1] without the curve changing in both cases?
  
  user73: Assuming you want a linear transformation (which is what "without the curve changing" sounds like to me), to rescale from range c..d to range 0..1 subtract the minimum then divide by the difference, i.e.
  $val_t = (val - c) / (d-c)$
  The inverse of this can be applied to work the other way, i.e.
  $inp_t = val \times (b-a) + a$
  For a function $f(x)$, with ranges as you have stated, the full transformation of input and output would be:
  $g(x_t) = \frac{f(x_t \times (b-a) + a) - c}{d-c}$
  
original_text: 14989: <p>When we have a function which takes values ​​in [a,b] and which returns values ​​in [c,d], what transformation can we do so that the function takes values ​​in [0,1] and returns values ​​in [0,1] without the curve changing in both cases?</p>
   73: <p>Assuming you want a linear transformation (which is what &quot;without the curve changing&quot; sounds like to me), to rescale from range <strong>c..d</strong> to range <strong>0..1</strong> subtract the minimum then divide by the difference, i.e.</p>
  <p><span class="math-container">$val_t = (val - c) / (d-c)$</span></p>
  <p>The inverse of this can be applied to work the other way, i.e.</p>
  <p><span class="math-container">$inp_t = val \times (b-a) + a$</span></p>
  <p>For a function <span class="math-container">$f(x)$</span>, with ranges as you have stated, the full transformation of input and output would be:</p>
  <p><span class="math-container">$g(x_t) = \frac{f(x_t \times (b-a) + a) - c}{d-c}$</span></p>
  