AP EAMCET · Maths · Functions
If a function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) is defined by \(f(x)=x-(-1)^x\), then \(f(x)\) is
- A one-one, but not onto
- B onto, but not one-one
- C both one-one and onto
- D neither one-one nor onto
Answer & Solution
Correct Answer
(C) both one-one and onto
Step-by-step Solution
Detailed explanation
To verify if \(f(x)\) is one-one: Assume \(f(a)=f(b) \Rightarrow a-(-1)^a=b-(-1)^b\). If \(a,b\) have same parity, then \((-1)^a=(-1)^b\), thus \(a=b\). If \(a\) is even and \(b\) is odd: \(a-1=b+1 \Rightarrow a-b=2\). This is impossible because \(a-b\) must be odd if \(a\) is…
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