A function f is given by a rule. Find the images and pre-images asked for, and identify the type of function.

1. 1 · Write what is given and what is to be found. Copy f, its domain and its co-domain.
2. 2 · Write the formula or theorem. One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂. Onto: range = co-domain.
3. 3 · Find the images. Substitute each x into the rule: f(1), f(2), f(3).
4. 4 · Find the pre-images. Put f(x) equal to the given value and solve for x. Check that x lies in the domain.
5. 5 · Test one-one. Assume f(x₁) = f(x₂) and show x₁ = x₂, or show two inputs with the same image.
6. 6 · Test onto. Name an element of the co-domain with no pre-image (into), or show that every y has one (onto).
7. 7 · Check and box the answer. Box the type in words, for example 'f is one-one and into'.

Scoring tip: For 'into', always give one element of the co-domain that has no pre-image. That one line carries a mark.