Many-to-One Mapping Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
The floor function f(x)=⌊x⌋f(x) = \lfloor x \rfloor maps every real number to the greatest integer ≤x\leq x. Show it is many-to-one and find f−1({3})f^{-1}(\{3\}).

Solution

  1. 1
    Many-to-one: all xx in [3,4)[3, 4) satisfy ⌊x⌋=3\lfloor x \rfloor = 3. For instance, f(3)=f(3.2)=f(3.99)=3f(3)=f(3.2)=f(3.99)=3 with distinct inputs.
  2. 2
    Pre-image of {3}\{3\}: f−1({3})={x:⌊x⌋=3}=[3,4)f^{-1}(\{3\}) = \{x : \lfloor x \rfloor = 3\} = [3, 4).
  3. 3
    This is an uncountably infinite set, illustrating how dramatically many-to-one a function can be.

Answer

ff is many-to-one; f−1({3})=[3,4)f^{-1}(\{3\}) = [3, 4)
The floor function collapses entire intervals to single integers. This is an extreme example of many-to-one behavior where infinitely many inputs share one output, which is why floor has no inverse function.

About Many-to-One Mapping

A many-to-one function maps multiple distinct inputs to the same output — it is a valid function (each input still has exactly one output) but has no inverse.

Learn more about Many-to-One Mapping →

More Many-to-One Mapping Examples

Example 1 easy

Show that [formula] is a many-to-one function by finding two distinct inputs that produce the same o

Example 3 easy

For [formula], find two values of [formula] in [formula] such that [formula].

Example 4 medium

For [formula], find all [formula] such that [formula], and explain why [formula] has no inverse on [

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