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Create a free account to open Conditional probability and independence. The free chapters of Probability stay readable without one.

Recipe

Test whether a function is one-to-one

Use when: you're given f: A \to B by a formula or an explicit list of pairs and asked whether it's one-to-one (injective)

Goal: either a general derivation showing f(a_1) = f(a_2) forces a_1 = a_2, or one explicit counterexample pair disproving it

Taught in One-to-one functions

Quick recipe

  1. Assume two outputs are equal and try to force the inputs equal
  2. Decide, from how the algebra goes, whether to finish the proof or hunt for a counterexample
  3. If hunting for a counterexample, test a small paired input
  4. State the final verdict with its justification
  1. 1

    Start by assuming f(a_1) = f(a_2) for two inputs, and see whether that assumption forces a_1 = a_2.

    Example

    For f(x) = 3x - 7: assume f(x_1) = f(x_2), so 3x_1 - 7 = 3x_2 - 7.

    Why: proving the contrapositive form, that equal outputs force equal inputs, is usually easier to work with algebraically than the original statement.

  2. 2

    Attempt to solve the equation from Step 1 for a_1 in terms of a_2, using only reversible operations like adding or dividing by a nonzero number.

    Example

    3x_1 - 7 = 3x_2 - 7: add 7 to both sides, giving 3x_1 = 3x_2, then divide by 3, giving x_1 = x_2. Every step here is reversible.

    • the algebra reduces cleanly to a1 = a2 — the function is one-to-one; write up the steps as the proof.
    • a step is not reversible, such as squaring, or you can already spot a repeated output — switch to hunting for a specific counterexample instead.

    Why: whether the algebra can be reversed determines whether the assumption truly forces the inputs equal, which is exactly what the definition demands.

  3. 3

    Try inputs that are natural candidates for colliding, such as a number and its negative, or two nearby integers, and compute both outputs.

    Example

    For f(n) = n^2: try n = -2 and n = 2. Both give f(-2) = 4 and f(2) = 4, even though -2 \ne 2.

    Why: one pair of distinct inputs with equal outputs disproves the property completely, so there is no need to check any other input.

  4. 4

    Write "one-to-one," backed by the completed derivation, or "not one-to-one," backed by the counterexample pair found.

    Why: the property is a claim about every possible pair of inputs, so the verdict is only trustworthy when it is tied to a general derivation or a specific counterexample.

Common mistakes

  • Testing only a couple of input pairs and calling the function one-to-one without a general derivation.
  • Squaring or taking an absolute value during the algebra, then treating that step as reversible when it isn't.
  • Finding a counterexample but reporting one-to-one anyway because most inputs behaved.
  • Checking that equal inputs give equal outputs (true of every function) instead of the reverse direction the definition actually requires.