I am a bit confused how the formula is different from a simple lerp.

    k(x) = 1/x
    
        y_i * k(x - x_i) + y_i+1 * k(x_i+1 - x)
    F = --------------------------------------- =
             k(x - x_i) + k(x_i+1 - x)
    
      y_i / (x - x_i) + y_i+1 / (x_i+1 - x)
    = --------------------------------------- =
           1 / (x - x_i) + 1 / (x_i+1 - x)
    
      y_i * (x_i+1 - x) / A + y_i+1 * (x - x_i) / A
    = ---------------------------------------------
           (x_i+1 - x) / A + (x - x_i) / A
    
    where A = (x - x_i) * (x_i+1 - x)
    scaling to x_i+1 - x_i = 1, and using a = (x - x_i)
    
    F = (1 - a) * y_i + a * y_i+1

The only time this wouldn't hold is when A = 0, but then F is not defined anyway. However, the graph of F shown is decidedly not that of the lerp function.