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One-to-one function

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College Algebra

Definition

A one-to-one function (injective function) is a function where each element of the domain maps to a unique element in the codomain. No two different elements in the domain map to the same element in the codomain.

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5 Must Know Facts For Your Next Test

  1. A function $f$ is one-to-one if and only if $f(a) = f(b)$ implies $a = b$ for all elements $a$ and $b$ in its domain.
  2. The horizontal line test can determine if a function is one-to-one: if every horizontal line intersects the graph at most once, then the function is one-to-one.
  3. One-to-one functions have inverses that are also functions.
  4. If a function is both one-to-one and onto, it is called bijective.
  5. In composition of functions, if $g \circ f$ is one-to-one, then $f$ must be one-to-one.

Review Questions

  • What property must hold true for all pairs of elements in the domain of a one-to-one function?
  • How can you use the horizontal line test to determine if a function is one-to-one?
  • Why do one-to-one functions always have inverses?
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