(a) Let f(x):(−∞,0)∪(0,∞)→R be defined by f(x)=x−1x. Show that f(x) has no inverse function. (b) Let g(x):(0,∞)→R be defined by g(x)=x−1x. Show that g(x) has an inverse function.
If you graph y = x - 1/x, then it clearly fails the Horizontal Line Test over all real numbers, so f is not invertible. But if you restrict to positive real numbers, then it passes the Horizontal LIne Test, so g is invertible.
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