Functions As Maps
Maps are a convenient way to visualise functions, or more generally, the association between two sets. A map relates one set to another using some rule. For example, suppose the rule is, \(y = f(x) = {x^2},\;{\rm{Set\,A}} = \{ 1,\;2,\;3,\;4\} ,\;{\rm{Set\,B}} = \{ 1,\;4,\;9,\;16\} \)
We can show this as a map.
The elements in \({\text{set B}}\) are the images of the elements of \({\text{set A}}\) under the action of \(f.\)
\({\text{Set A}}\) would be called the domain of \(f\left\{ {{\text{the set of all input values}}}\right\},\) and \({\text{set B}}\) the co-domain of
\(f\left\{ {{\text{the set in which the output values lie}}}\right\}.\) Here, the co-domain is the same as the range{precisely
the set of all outputs}, but in general, range would be a subset of the co-domain
Range \( \subseteq \) co-domain.
For our purposes, co-domain is not of much significance. It is more of a matter of convenience to specify the co-domain.
Suppose I want to talk about \(f(x) = \sqrt {a - x} + \sqrt {x - b} ,\,a > b > 0\) (the function we analysed earlier). Note that the domain is \([b,\,a].\) The output will obviously be real and positive (This much I can tell just by looking at the definition of \(f(x)\)).
Hence, I could specify the definition of \(f\) as
As we can see above, co-domain just tells us that all the outputs will lie in it. The exact set of outputs (the range) will lie in the co-domain, and will be its subset.
We know that the range for this \(f(x)\) is \([\sqrt {a - b} ,\sqrt {2(a - b)} ]\)
Therefore, I can also specify \(f\) as
In this definition, co-domain is the same as range. Note that co-domain can be at the most equal to range (not smaller).
The map in the figure above is an example of a one-one and onto map. Let us see what this means, and what are the various possible kinds of maps.
(A) ONE-ONE / MANY – ONE
One-one implies that each element in the domain set is mapped to only one element in the co-domain set, that is, no two inputs map to the same output.
The one-one condition can be written mathematically as
\(f({x_1}) = f({x_2})\) if and only if \({x_1} = {x_2}\)
This, stated in words, says that two outputs from f can be equal if and only if the corresponding inputs to f are equal.
If \(f({x_1}) = f({x_2})\) does not imply \({x_1} = {x_2}\), then the function is many-one. Therefore, given \(f,\) we can find whether its one-one or not by solving the equation \(f({x_1}) = f({x_2})\). If this equation yields \({x_1} = {x_2}\), then \(f\) is one-one else not. As an example, consider \(f(x)=x^2.\) Is this one-one, or many one?
Solving
\(f({x_1}) = f({x_2}) \qquad \Rightarrow \qquad x_1^2 = x_2^2\)
\(\begin{align}&\Rightarrow \qquad ({x_1} + {x_2})({x_1} - {x_2}) = 0\\& \Rightarrow \qquad {x_1} = {x_2}\,\,\,{\rm{or}}\,\,\,{x_1} = - {x_2} \end{align}\)
Hence \(f({x_1}) = f({x_2})\) does not uniquely imply \({x_1} = {x_2}\) (Because another case exists, \({x_1} = -{x_2}\) )
For example \(f(3) = f( - 3) = 9\) (One output for two different inputs). Hence \(f(x) = {x^2}\) is many-one
On a graph, this implies that if we draw any horizontal line, and it intersects the graph at the most once, then it is one-one, else it is many-one.
Note: One-many or many-many maps also exist but they are not functions. A function implies that each element in the domain is associated with one and only one element in the co-domain
For the graph of a valid function, this implies that if we draw a vertical line, it should intersect the graph at the most once. Otherwise, it is not a function.
(B) INTO/ONTO
Look carefully at the three maps below. The first and the third are many-one, while the second is one-many.
Any other differences? For the first two maps, each element in the co-domain (\({\text{Set B}}\)) is ‘covered’, that is, associated with some element in the domain (\({\text{Set A.}}\)) For the third map, one element \((\{ f\})\) in the co-domain is ‘left out’ (is not associated with any element in the domain, or in other words, does not have a pre-image). The first two maps are onto while the third is into.
Stating formally, an onto map is a map in which the range (the set of all images of the elements of the domain) ‘covers’ the entire co-domain, that is, \({\bf{\text{Range}}}={\bf{\text{Co-domain}}}.\)
If \({\text{Range}}\subset{\text{Co-domain}}\) (like the third map above), then it is an into map.
An example will make all this more clear:
The range is the set of all non-negative numbers, as \({x^2} \ge 0.\) Hence, Range \( \subset \) Co-domain and the function is into. On the other hand, for
Range = Co-domain and the map is onto.