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