Sets and mappings
Objective. Use set operations, images, inverse images and Cartesian products.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Sets collect objects, and mappings (functions) assign to each element of a domain exactly one element of a codomain, written f:A→B. The image of a subset S⊆A is f(S)={f(x):x∈S}, while the inverse image of T⊆B is f⁻¹(T)={x∈A:f(x)∈T}, and this works even when f has no inverse function. The Cartesian product A×B={(a,b):a∈A,b∈B} underlies graphs of functions and relations. For f:ℝ→ℝ, f(x)=x², the image of [−1,2] is [0,4], and the inverse image of {4} is {−2,2}. Distinguishing f(S) from f⁻¹(T) and remembering inverse images always exist (as sets) are core skills for later analysis.
Worked example
For the mapping f:ℝ→ℝ given by f(x)=x²−1, find the inverse image f⁻¹({3}).
- 1We need all x with f(x)=3, i.e. x²−1=3.
- 2Rearrange to x²=4.
- 3Take square roots: x=2 or x=−2.
- 4Check: f(2)=4−1=3 and f(−2)=4−1=3, both valid, so f⁻¹({3})={−2,2}.
Answer: {−2, 2}
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Practice
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Start practisingCommon mistake. Students assume f⁻¹(T) requires f to be invertible, writing it only when f⁻¹ exists as a function. In fact f⁻¹(T) is simply the set of preimages and is always defined, even for non-injective f.
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