The function f(x) is defined for all real x. If `f(a+b)=f(ab) AA a " and " b " and " f(-(1)/(2))=-(1)/(2)` then find the value of `f(1005).`

Explanation

We have `(f(a+b)=f(ab) AA a " and " b`
Put ` a=0`
` :. f(b)=f(0)=k ` (say)
Putting `b=0` in given relation,
`f(a)=f(0)=k`
Thus `f(a)=f(b)=k AA a,b`
Hence, `f(x)` is a constant function.
` :. f(1005)=f(-(1)/(2))`
`=-(1)/(2)`