f(a) ≠ 0 for some a in R
f(xf(y)) = yf(x) for all x,y in R
First we find the value of f(0)
f(0) = f(0f(0)) = 0f(0) = 0 (1)
Then a property of f
f(w) = f(z)
==> wf(a) = f(af(w)) = f(af(z)) = zf(a)
==> w = z
Therefore, f is one-to-one (or injective). (2)
Using (2) we have
z ≠ 0
==> f(zf(xy)) = xyf(z) = xf(zf(y)) = f(zf(y)f(x))
==> zf(xy) = zf(x)f(y)
==> f(xy) = f(x)f(y) (3)
Using (1), (2), and (3) we have
x = y = 1
==> f(1)^2 = f(1)f(1) = f(1*1) = f(1)
==> f(1) = 0 or 1
==> f(1) = 1 (4)
Using (2), (3), and (4) we have
x = y = -1
==> f(-1)^2 = f(-1)f(-1) = f((-1)*(-1)) = f(1) = 1
==> f(-1) = 1 or -1
==> f(-1) = -1 (5)
Finally, using (3) and (5) we have
f(-x) = f(-1x) = f(-1)f(x) = -1f(x) = -f(x)
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