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Mathematics LibreTexts

6.S: Functions (Summary)

  • Page ID
    7073
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    Important Definitions

    • Function, page 284
    • Domain of a function, page 285
    • Codomain of a function,page285
    • Image of \(x\) under \(f\), page 285
    • preimage of \(y\) under \(f\), page 285
    • Independent variable, page 285
    • Dependent variable, page 285
    • Range of a function, page 287
    • Image of a function, page 287
    • Equal functions, page 298
    • Sequence, page 301
    • Injection, page 310
    • One-to-one function, page 310
    • Surjection, page 311
    • Onto function, page 311
    • Bijection, page 312
    • One-to-one and onto, page 312
    • Composition of \(f\) and \(g\), page 325
    • Composite function, page 325
    • \(f\) followed by \(g\), page 325
    • Inverse of a function, page 338
    • Image of a set under a function, page 351
    • preimage of a set under a function, page 351

    Important Theorems and Results about Functions

    • Theorem 6.20. Let \(A\), \(B\) and \(C\) be nonempty sets and let \(f: A \to B\) and \(g: B \to C\).

      1. If \(f\) and \(g\) are both injections, then \(g \circ f\) is an injection.
      2. If \(f\) and \(g\) are both surjections, then \(g \circ f\) is a surjection.
      3. If \(f\) and \(g\) are both bijections, then \(g \circ f\) is a bijection.
    • Theorem 6.21. Let \(A\), \(B\) and \(C\) be nonempty sets and let \(f: A \to B\) and \(g: B \to C\).

      1. If \(g \circ f: A \to C\) is an injection, then \(f: A \to B\) is an injeciton.
      2. If \(g \circ f: A \to C\) is a surjection, then \(g: B \to C\) is a surjeciton.
    • Theorem 6.22. Let \(A\) and \(B\) be nonempty sets and let \(f\) be a subset of \(A \times B\) that satisfies the following two properties:

      \(\bullet\) For every \(a \in A\), there exists \(b \in B\) such that \((a, b) \in f\); and
      \(\bullet\) For every \(a \in A\) and every \(b, c \in B\), if \((a, b) \in f\) and \((a, c) \in f\), then \(b = c\).

      If we use \(f(a) = b\) whenever \((a, b) \in f\), then \(f\) is a function from \(A\) to \(B\).
    • Theorem 6.25. Let \(A\) and \(B\) be nonempty sets and let \(f: A \to B\). The inverse of \(f\) is a function from \(B\) to \(A\) if and only if \(f\) is a bijection.
    • Theorem 6.26. Let \(A\) and \(B\) be nonempty sets and let \(f: A \to B\) be a bijection. Then \(f^{-1}: B \to A\) is a function, and for every \(a \in A\) and \(b \in B\),
      \(f(a) = b\) if and only if \(f^{-1}(b) = a\).
    • Corollary 6.28. Let \(A\) and \(B\) be nonempty sets and let \(f: A \to B\) be a bijection. Then

      1. For every \(x\) in \(A\), \((f^{-1} \circ f)(x) = x\).
      2. For every \(y\) in \(B\), \((f \circ f^{-1} (y) = y\).
    • Theorem 6.29. Let \(f: A \to B\) and \(g: B \to C\) be bijections. Then \(g \circ f\) is a bijection and \((g \circ f)^{-1} = f^{-1} \circ g^{-1}\).
    • Theorem 6.34. Let \(f: S \to T\) be a function and let \(A\) and \(B\) be subsets of \(S\). Then

      1. \(f(A \cap B) \subseteq f(A) \cap f(B)\)
      2. \(f(A \cup B) = f(A) \cup f(B)\)
    • Theorem 6.35. Let \(f: S \to T\) be a function and let \(C\) and \(D\) be subsets of \(T\). Then

      1. \(f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)\)
      2. \(f^{-1}(C \cup D) = f^{-1}(C) \cap f^{-1}(D)\)
    • Theorem 6.36. Let \(f: S \to T\) be a function and let \(A\( be a subset of \(S\) and let \(C\) be a subset of \(T\). Then

      1. \(A \subseteq f^{-1}(f(A))\)
      2. \(f(f^{-1}(C) \subseteq C\)