Session 3C
Functions: injective, surjective, bijective
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- A function f: A -> B assigns each input in A exactly one output in B. The set A is the domain and B is the codomain. A function is injective (one-to-one) if different inputs can't map to the same output. The standard proof format is: assume f(a) = f(b) and then show a = b. A function is surjective (onto) if every element of the codomain is hit: for each y in B, there exists x in A such that f(x) = y. The proof format is constructive: start with an arbitrary y, then solve f(x)=y for x and show that x lies in the domain.
- A function is bijective if it is both injective and surjective. Bijectivity matters because bijections are exactly the functions that have inverses. Many beginner mistakes come from mixing up domain and codomain. For example, f(x)=x^2 from R to R is not surjective because negative numbers are not hit, but from R to [0, infinity) it is surjective. Function proofs are often algebra plus careful attention to where variables live. The point is not complicated computation. It's demonstrating logical control over definitions.
Worked examples
- f(x) = 2x + 1 from Z to Z is injective.
Practice
- 1Show f(x) = x^2 from R to R is not injective.
- 2Show f(x) = x^3 from R to R is bijective.
- 3Determine if f(x) = 2x from Z to Z is surjective.
Show answers / outlines
Answers
- 1f(1) = f(-1).
- 2Injective: a^3 = b^3 implies a = b; surjective: x = cbrt(y).
- 3Not surjective: odd integers are not hit.
MCQ
The function f(x) = x^2 from R to R is: