The function $f$ that maps the set of all natural numbers onto the set of all integers by multiplying each element of the domain by minus one can be written in symbolic notation as:

$f: \mathbb{N} \to \mathbb{Z}, \quad f(n) = -n \quad \text{for all} \, n \in \mathbb{N}$

Here, $\mathbb{N}$ represents the set of natural numbers, and $\mathbb{Z}$ represents the set of integers.

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the difference between surjection and injection for functions?
2. Can the set of natural numbers be mapped one-to-one onto the set of integers?
3. How would you represent a function mapping $\mathbb{Z}$ to $\mathbb{Z}$ with a different transformation?
4. How does the codomain affect the properties of a function?
5. What are the key differences between the sets $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{R}$?

Tip: A function is said to be onto (surjective) if every element in the codomain is the image of at least one element from the domain.