Lets start Preparing hard for KVPY SX!(1)

If a function $f : R \rightarrow R$ is given by the functional equation,

$\large{f(x-f(y))=f(f(y))+xf(y)+f(x)-1}$ .

Now let's take the function $f(x)$ where $x$ is obviously a real number as per our definition of the function.

• If the function is injective, take $z=2$
• If the function is onto, take $z=3$
• If the function is into, take $z=1$ .

If the value of

$\displaystyle{\int_{0}^{|z|} f^{-1}(t). dt = \dfrac{a^{{m/n}}}{b}}$

Here, a is a prime number. And gcd(a,b)=1 . ||Find the value of $f(1)+f(2)+a+b+m+n+|z|$ .

Details and Assumptions

• Here $|\cdot|$ denotes absolute value function.
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