Let $\mathbb { N }$ denote the set of all natural numbers, and $\mathbb { Z }$ denote the set of all integers. Consider the functions $f : \mathbb { N } \rightarrow \mathbb { Z }$ and $g : \mathbb { Z } \rightarrow \mathbb { N }$ defined by

$$f ( n ) = \begin{cases} ( n + 1 ) / 2 & \text { if } n \text { is odd } \\ ( 4 - n ) / 2 & \text { if } n \text { is even } \end{cases}$$

and

$$g ( n ) = \begin{cases} 3 + 2 n & \text { if } n \geq 0 \\ - 2 n & \text { if } n < 0 \end{cases}$$

Define $( g \circ f ) ( n ) = g ( f ( n ) )$ for all $n \in \mathbb { N }$, and $( f \circ g ) ( n ) = f ( g ( n ) )$ for all $n \in \mathbb { Z }$.

Then which of the following statements is (are) TRUE?

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\begin{tabular}{|l|l|}
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(A) & $g \circ f$ is NOT one-one and $g \circ f$ is NOT onto \\
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(B) & $f \circ g$ is NOT one-one but $f \circ g$ is onto \\
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(C) & $g$ is one-one and $g$ is onto \\
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(D) & $f$ is NOT one-one but $f$ is onto \\
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\end{tabular}
\end{center}