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  4. Exactly how many function f: Q arrow Q exist such that f(x+ y)=f(x)+f(y) and f(x y)=f(x) f(y) for all x, y ∈ Q ?

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Q. Exactly how many function $f: Q \rightarrow Q$ exist such that $f(x+ y)=f(x)+f(y)$ and $f(x y)=f(x) f(y)$ for all $x, y \in Q$ ?

AP EAMCETAP EAMCET 2020

Solution:

These functions are simultaneously satisfy for $f(x)=x$ and $f(x)=0 \quad \forall x \in Q$