Question

The table shows the orbital period of the eight major planets in our Solar System given their mean distance from the Sun. Define the domain and range of the function and state whether it is one-to-one, onto, both or neither.



Planet Mean Distance from Sun (AU) Orbital Period (years)
Mercury 0.4 0.241
Venus 0.7 0.615
Earth 1.0 1.0
Mars 1.5 1.881
Jupiter 5.2 11.75
Saturn 9.5 29.5
Uranus 19.2 84
Neptune 30 164.8

Answer 1

Interpret the function f as mapping mean distance (AU) → orbital period (years).

- Domain = {0.4, 0.7, 1.0, 1.5, 5.2, 9.5, 19.2, 30}.
- Range = {0.241, 0.615, 1.0, 1.881, 11.75, 29.5, 84, 164.8}.

Is it one-to-one? Yes. Each distance maps to a distinct orbital period, so f is injective.

Is it onto? Not if the codomain is taken to be all real numbers (or all possible years): the function does not produce every real number, so it is not surjective. (If you declare the codomain to be exactly the set listed above as the range, then f is onto that codomain and hence bijective.)