Math Revision 16.2 - Injective, Surjective and Bijective Functions. Graphs of Functions

In these revision notes for Injective, Surjective and Bijective Functions. Graphs of Functions, we cover the following key points:

• What is codomain? Where does it differ from the range?
• What are injective functions?
• What are surjective functions?
• What is the condition for a function to be bijective?
• How to build the graph of a function?
• What is the horizontal line test? What is it is used for?
• What is the vertical line test? What is it is used for

Injective, Surjective and Bijective Functions. Graphs of Functions Revision Notes

The domain D is the set of all values the independent variable (input) of a function takes, while range R is the set of the output values resulting from the operations made with input values. Another concept encountered when dealing with functions is the Codomain Y. It includes all possible values the output set contains. Hence, the Range is a subset of (is included in) the Codomain.

Based on the relationship between variables, functions are classified into three main categories (types). The first type of function is called injective; it is a kind of function in which each element of the input set X is related to a distinct element of the output set Y. It is a kind of one-to-one function, but where not all elements of the output set are connected to those of the input set. The formal definition of injective function is as follows:

"A function f is injective only if for any f(x) = f(y) there is x = y."

The second type of function includes what we call surjective functions. In such functions, each element of the output set Y has in correspondence at least one element of the input set X. In other words, unlike in injective functions, in surjective functions, there are no free elements in the output set Y; all y-elements are related to at least one x-element. In other words, in surjective functions, we may have more than one x-value corresponding to the same y-value.

The formal definition of surjective functions is as below:

"A function f (from the input set X to the output set Y) is surjective only if for every y in Y, there is at least one x in X such that f(x) = y. In other words, the function f(x) is surjective only if f(X) = Y."

The third type of function includes what we call bijective functions. By definition, a bijective function is a type of function that is injective and surjective at the same time. In other words, a surjective function must be one-to-one and have all output values connected to a single input.

For example, all linear functions defined in R are bijective because every y-value has a unique x-value in correspondence. We can define a bijective function in a more formal language as follows:

"A function f(x) (from set X to Y) is bijective if, for every y in Y, there is exactly one x in X such that f(x) = y."

Some functions may be bijective in one domain set and bijective in another.

The graph of a function is a geometrical representation of the set of all points (ordered pairs) which - when substituted in the function's formula - make this function true.

In general, for every numerical function f: X → R, the graph is composed of an infinite set of real ordered pairs (x, y), where x ∊ R and y ∊ R. Every such ordered pair has in correspondence a single point in the coordinates system XOY, where the first number of the ordered pair corresponds to the x-coordinate (abscissa) of the graph while the second number corresponds to the y-coordinate (ordinate) of the graph in that point.

The horizontal line test is a method used to check whether a function is injective (one-to-one) or not when the graph of the function is given. It consists of drawing a horizontal line in doubtful places to 'catch' any double intercept of the line with the graph. In that case, there is a single y-value for two different x-values - a thing which makes the given function unqualifiable for being injective and therefore, bijective. Therefore, such a function can be only surjective but not injective.

A good method to check whether a given graph represents a function or not is to draw a vertical line in the sections where you have doubts that an x-value may have in correspondence two or more y-values. If the vertical line intercepts the graph at more than one point, that graph does not represent a function.

This feature which allows us to check whether a graph belongs to a function or not, is called the "vertical line test." We have established that not all relations are functions, therefore, since every relation between two quantities x and y can be mapped on the XOY coordinates system, the same x-value may have in correspondence two different y-values. This results in points that when shown in a graph, lie in the same horizontal position (the same x-coordinate) but at two different heights (different y-coordinates).