Let . If we can establish a full bijection between f and , then must take all real values. Let's see, say :
The square root is bothersome, as we need to show that the value under is (hopefully) always positive. So :
Discriminant is . Since the quadratic coefficient, , is positive, this quadratic yields only positive values.
Therefore the equation :
Does accept all values of . And therefore, in :
does take all real values. does take all real values for . QED.