Types of Functions Review

Table of Contents

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:

(1)

\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:

(2)

\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 $\sigma : S \to S$ .

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