If f(0) = 1 and f(2xy - 1) = f(x)f(y) - f(x) - 2y – 1, then f(x) = ?
Setting x = 0 in the expression f(2xy - 1) = f(x)f(y) - f(x) - 2y – 1, then,
f(0 - 1) = f(0)f(y) - f(0) - 2y -1
f(-1) = f(y) -1 - 2y - 1
f(-1) = f(y) - 2y - 2
or f(y) = f(-1) + 2y + 2
setting y = 0 in the above equation,
f(0) = f(-1) + 0 + 2
f(-1) = 1 - 2 = -1
f(-1) = -1
Using this value for f(-1), we can rewrite the equation for f(y), to give
f(y) = -1 + 2y + 2
f(y) = 2y + 1
Therefore f(x) = 2x + 1