Monday, March 19, 2007
A couple of algebra class puzzles
I've always wondered how you get from the usual definition of a hyperbola ((x-h)**2 / a**2) - (y-k) ** 2 / b **2 ) and variations (opening East-west or north-south -- see Wkipedia here) to the "gas law" like graph xy = k that has asymptotes along the x and y axes or parallel lines. There aren't a lot of sites that explain the derivation, but here is one.
Another teaser question in algebra has to do with inverse functions. (Wiki reference) f(x) and g(x) are inverses if f[g(x)] = g[f(x)] = x. (x, not 1). So what is the inverse of f(x) = x? It is still g(x) = x. It's not 1/x. If you plug that in, you get 1.