6.S: Functions (Summary)

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