6.S: Functions (Summary)

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

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