Today is the day I knew, beyond a shadow of a doubt, that my decision to finally outsource my son's math instruction this year was a good one. This is the letter to his teacher--written totally "in Greek"--which I found cc'ed in my inbox today:
When we find the inverse of a function and then prove that f(f-1(x)) = x and f-1(f(x)) = x, does that mean that f(x) = f-1(x)?
If we solve for f(x) = f-1(x), though, they don't cancel each other out. For instance:
f(x) = 6x
f-1(x) = x/6
If f(x) = f-1(x), then 6x = x/6, which is impossible, because then:
6x = x/6
x = x/36
1 = x/36x
1 = 1/36
x/6 = 6x
x = 36x
1 = 36x/x
1 = 36
But, since f(f-1(x)) = x is true, f-1(f(x)) = x is true, and f(x) ≠ f-1(x) is true, does that mean that x ≠ x? I'm a little confused by this... Could you explain it for me?
Hunh?! Thankfully, he is now asking somebody besides me!!