The Day I Knew...

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:

Mrs. M.,
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

OR

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?

Thanks,
PT

Hunh?! Thankfully, he is now asking somebody besides me!!