CUET · MATHS · PYQ PAPER 2023
Consider the function \(f: N \rightarrow N\) given by \(f(x)=\left\{\begin{array}{ll}x+1, & \text { if } x \text { is odd } \\ x-1, & \text { if } x \text { is even }\end{array}\right.\)
- A f is neither one-one nor onto
- B f is one-one but not onto
- C f is onto but not one-one
- D f is both one-one and onto
Answer & Solution
Correct Answer
(D) f is both one-one and onto
Step-by-step Solution
Detailed explanation
One-one: Assume \(f(x_1) = f(x_2)\). If \(x_1\) is odd, \(f(x_1) = x_1+1\) (even). If \(x_2\) is even, \(f(x_2) = x_2-1\) (odd). These cannot be equal. Thus \(x_1, x_2\) must be of the same parity. If \(x_1, x_2\) both odd: \(x_1+1 = x_2+1 \implies x_1 = x_2\). If \(x_1, x_2\)…
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