TS EAMCET · Maths · Functions
The number of bijective functions \(f: \mathbf{Z} \rightarrow \mathbf{Z}\) such that \(f(x+y)=f(x)+f(y) \forall x, y \in \mathbf{Z}\), is
- A two
- B four
- C zero
- D infinitely many
Answer & Solution
Correct Answer
(A) two
Step-by-step Solution
Detailed explanation
Let \(x\) and \(y\) be any two elements in the domain \((Z)\), such that \(f(x+y)=f(x)+f(y)\) \(\ldots\) (i) Differentiating above expression w.r.t ' \(y\) ', keeping \(x\) constant, we get \(f^{\prime}(x+y)=f^{\prime}(y)\) Let \(y=0 \Rightarrow f^{\prime}(x+0)=f^{\prime}(0)\)…
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