Question

Suppose that \(f(x)\) is a function such that \(f(xy) + x = xf(y) + f(x)\) for all real numbers \(x\) and \(y\). If \(f(-1) = 5\) then compute \(f(-1001)\).

Answer 1

\(f(-1 \times a) = f(a \times -1) = f(-a) \)

x=-1, y = a
\(\implies f(-a) -1 = -f(a) + f(-1)\)
\(f(-a) = -f(a) + 6\)

x=a, y = -1
\(\implies f(-a) + a = af(-1) + f(a)\)
\(f(-a) = f(a) + 4a\)

\(f(a) = 3 - 2a\)
\(f(-a) = 3 + 2a\)

2005 either way