Verify that the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f(x) = 2x - 7\) is one-one but not onto.
Proof:
To check the function is one-to-one, we assume .
For , .
Thus, \(f\) is one-to-one.
To check the function is onto, let such that \(y \in \mathbb{N}\).
Then, .
If \(y = 2\):
\(x = \frac{2 + 7}{2} = \frac{9}{2} = 4.5\)
This is not possible because \(x \in \mathbb{N}\).
Thus, \(f\) is not onto.
Hence, we proved.
Answer variants:
\(f(x_1) = f(x_2)\)
\(f(x) = y\)
\(x_1 = x_2\)