Types of Functions Review

# Types of Functions Review

We will now review some of the recent material regarding types of functions.

- On the
**Injective, Surjective, and Bijective Functions**page we said that a**Function**from a set $A$ called the**Domain**to a set $B$ called the**Codomain**denoted $f : A \to B$ is a rule which maps every element $x \in A$ to exactly one element $y \in B$ which we denote $f(x) = y$. We said that the element $y$ is the**Image**of $x$ under $f$. We defined the set of all elements in $B$ that are the image of some element in $A$ under $f$ to be the**Range**of $f$.

- We said that a function $f : A \to B$ is
**Injective**or**One-to-one**if whenever $x, y \in B$ are such that $f(x) = f(y)$ we have that $x = y$.

- We said that a function $f : A \to B$ is
**Surjective**or**Onto**if the range of $f$ is all of $B$.

- We said that a function $f : A \to B$ is
**Bijective**if $f$ is both injective and surjective.

- On
**The Composition of Two Functions**page we said that if $f : A \to B$ and $g : B \to C$ then the**Composition Function**denoted $g \circ f : A \to C$ is defined for all $x \in A$ by:

\begin{align} \quad (g \circ f) (x) = g(f(x)) \end{align}

- We looked at many properties of compositions of two functions which are summarized below:

Property |
---|

(a) If $f : A \to B$ and $g : B \to C$ are injective then $g \circ f$ is injective. |

(b) If $f : A \to B$ and $g : B \to C$ are surjective then $g \circ f$ is surjective. |

(c) If $f : A \to B$ and $g : B \to C$ are bijective then $g \circ f$ is bijective. |

- We then looked at some more results on the
**Basic Theorems on the Composition of Two Functions**page which are summarized below:

Property |
---|

(a) If $f : A \to B$, $g : B \to C$, and if $g \circ f : A \to C$ is injective, then $f$ is injective. |

(b) If $f : A \to B$, $g : B \to C$, and if $g \circ f : A \to C$ is surjective, then $g$ is surjective. |

(c) If $f : A \to B$, $g : B \to C$, and if $g \circ f : A \to C$ is bijective, then $f$ is injective and $g$ is surjective. |

- On
**The Inverse of a Function**page we saw that if $f : A \to B$ is a bijective function, we can define the**Inverse Function**of $f$ denoted $f^{-1} : B \to A$ as follows. For each $x \in A$ we have that $f(x) = y$. So $f^{-1}(y) = x$. With this definition we have that for all $x \in A$ and for all $y \in B$ that:

\begin{align} \quad f^{-1}(f(x)) = x \quad \mathrm{and} \quad f(f^{-1}(y)) = y \end{align}

- On
**The Identity Function**page we defined a special function. We said that the**Identity Function**on the set $A$ denoted $i : A \to A$ is defined for all $x \in A$ by $i(x) = x$. The identity function on a set $A$ has many properties that are easy to acknowledge such that $i$ is bijective and $i^{-1} = i$.

- On the
**Permutations of Elements in a Set as Functions**we defined a special type of function. We said that a**Permutation**on a set $S$ is simply a bijective function $\sigma : S \to S$.