Practice question · Multiple choice
A function must assign exactly one output to each input. Why does the "exactly one" requirement do more work than the "at least one" part?
Hints
- Suppose f(3) could be either 5 or 7. What does the expression f(3) + 1 equal?
- Ask what has to be true before you can substitute f(x) into another expression.
Show the answer
B. Because it is what makes f(x) denote a determinate object.
Why
Uniqueness is what turns the notation into a name: if f(3) could be 5 or 7 then f(3) + 1 has no value and nothing built on f can be reasoned about. It is also why an inverse need not be a function, reversing y = x² sends 4 to both 2 and −2, which is exactly the situation ruled out, and why inverses require injectivity.
Practise Functions as Mappings
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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?
- Order the steps of a proof that f(n) = 3n - 2 is injective.
- 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.