Practice question · Put in order
Order the steps of a proof that f(n) = 3n - 2 is injective.
- Suppose f(a) = f(b) for some integers a and b
- Then 3a - 2 = 3b - 2
- Divide by 3: a = b
- Add 2 to both sides: 3a = 3b
- Since equal outputs force equal inputs, f is injective
Hints
- Injectivity proofs start by assuming two outputs agree.
- The goal is to force the inputs to be equal.
Show the answer
- Suppose f(a) = f(b) for some integers a and b
- Then 3a - 2 = 3b - 2
- Add 2 to both sides: 3a = 3b
- Divide by 3: a = b
- Since equal outputs force equal inputs, f is injective
Why
The standard template: assume f(a) = f(b), unwind the rule algebraically, and conclude a = b. Reaching a = b is exactly the definition of injective, so the proof ends there.
Practise Functions as Mappings
The app has 5 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Functions as Mappings
- Let f have domain {-2, -1, 0, 1, 2} and rule f(n) = n squared. How many distinct elements are in its image?
- Let f(n) = 2n with domain and codomain the integers. Select every integer that IS in the image of f.
- The rule "send x to its square root" is not a function on the reals, but √x is. What was fixed?
- A function must assign exactly one output to each input. Why does the "exactly one" requirement do more work…
- Every function from the integers to the integers is surjective.
- A function assigns exactly one output to each input. Sort each correspondence.
- Complete the definition of an injective function.