Question 1. Consider the formula: $f(x) = \frac{1}{x}$.
For each choice of domain X and codomain Y listed below, determine if the formula given above defines a function $f: X \to Y$. Explain your reasoning.
Moreover, exactly one pair X, Y below yields a bijective function. Identify this pair and prove that the corresponding function is a bijection by proving it is both injective and surjective.
1. $X = \mathbb{R}$, $Y = \mathbb{R}$
2. $X = \mathbb{N} \setminus \{0\}$, $Y = \mathbb{N}$
3. $X = \mathbb{N} \setminus \{0\}$, $Y = \mathbb{Q}$
4. $X = \mathbb{R} \setminus \{0\}$, $Y = \mathbb{R} \setminus \{0\}$
5. $X = \mathbb{R}$, $Y = \mathbb{N} \setminus \{0\}$
