The function

*f*satisfies the functional equation*f*(

*x*) +

*f*(

*y*) =

*f*(

*x*+

*y*) –

*xy*– 1

for every pair

*x*,*y*of real numbers. If*f*(1) = 1, then the number of positive integers*n*for which*f*(*n*) =*n*is:- 0
- 1
- 2
- 3
- infinite

We are given *f*(1) = 1. Let’s find the functional values of some other integers.

*f*(

*x*) +

*f*(

*y*) =

*f*(

*x*+

*y*) –

*xy*– 1

*f*(

*x*) +

*f*(0) =

*f*(

*x*+ 0) –

*x*· 0 – 1 =

*f*(

*x*) – 1

So

*f*(0) = -1

*f*(1) + *f*(1) = *f*(2) – 1 · 1 – 1

1 + 1 = *f*(2) – 1 – 1

And *f*(2) = 4

*f*(2) + *f*(1) = *f*(3) – 2 · 1 – 1

4 + 1 = *f*(3) – 2 – 1

Then *f*(3) = 8

Let’s make a chart and look for a pattern.

x |
f(x) |

0 | -1 |

1 | 1 = f(0) + 2 |

2 | 4 = f(1) + 3 |

3 | 8 = f(2) + 4 |

4 | 13 = f(3) + 5 |

… | |

n |
= f(n – 1) + n + 1 |

So, for

*f*(*n*) =*n*, we have to have: *f*(*n*) = *f*(*n* – 1) + *n* + 1 = *n*, so

*f*(*n* – 1) + 1 = 0, and

*f*(*n* – 1) = -1.

We see that *n* = 1 works and no other positive integers, *n* > 1, will work. So the answer is (b).