Finally, in a recent episode of the podcast Talkin’ Tunes with Rob Paulsen , guest Billy West sat down to recreate a scene from Back to the Future , substituting his Futurama characters Fry and Professor Farnsworth for Marty McFly and Doc Brown. The scene in question is when Marty finds Doc after traveling back to 1955, and it’s incredible to watch Billy West go back and forth between the voices. The voices don’t match what West looks like, so it never appears as if he’s actually the one doing the speaking. It’s all so bizarre and impressive.
To construct a bijection from T to R , start with the tangent function tan( x ), which is a bijection from (−π/2, π/2) to R (see the figure shown on the right). Next observe that the linear function h ( x ) = π x – π/2 is a bijection from (0, 1) to (−π/2, π/2) (see the figure shown on the left). The composite function tan( h ( x )) = tan(π x – π/2) is a bijection from (0, 1) to R . Composing this function with g ( t ) produces the function tan( h ( g ( t ))) = tan(π g ( t ) – π/2) , which is a bijection from T to R .