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 $\sigma : S \to S$.
