When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and co-domain.
The “domain” of a function or relation is:
- the set of all values for which it can be evaluated
- the set of allowable “input” values
- the values along the horizontal axis for which a point can be plotted along the vertical axis
For example, the following functions can be evaluated for any value of “x”:
therefore their domains will be “the set of all real numbers”.
The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:
The values for which a function or relation cannot be evaluated are called “domain restrictions“, and they usually arise either because:
- The person defining the function wished to exclude them, such as when creating a piecewise function. The function k(x) above is an example of this situation.
- The result of the function would be “undefined” for that value, such as when a value of “x” would require us to divide by zero at some point while evaluating the function. Functions h(x) and j(x) above are examples of this situation.
When graphing a function or relation, domain restrictions will result in a “hole” in the graph: one or more values on the horizontal axis for which no point can be graphed along the vertical axis.
Some holes in a graph correspond to the location of an “asymptote“: a line that the graph gradually approaches, and gets infinitely close to, but never crosses. If you examine the graphs of h(x) and j(x) as defined above, they will both exhibit this behavior at their domain restrictions, while k(x) does not.
The “range” of a function or relation is:
- the set of all values that it can produce
- its “output” set of values
- the set of values along the vertical axis for which a point can be plotted on its graph
Range restrictions usually occur due to the nature of the function or the relation… they are not usually imposed by the author, as is often done with domain restrictions (such as in k(x) above). For example:
- f(x) above does not have any restrictions to its Range… it can produce all real numbers. Therefore its Range is “the set of all Real numbers”.
- g(x) above does not produce a real number less than 5 (since x squared can never be less than zero), therefore the Range of g(x) is “all real numbers greater than or equal to five”.
Note that the word “range” has a number of possible uses in the English language, many of which do NOT refer to “the range of a function”. So, please read questions carefully to determine what the word “range” refers to in the question. If you are being asked for the “range of a function” – the above description applies. However, if you are being asked for the “range of possible values for a coefficient”, then you are NOT being asked for the “range” as defined above.
The “co-domain” of a function or relation is a set of values that includes the Range as described above, but may also include additional values beyond those in the range.
Co-domains can be useful when:
- You need to restrict the output of a function. For example, by specifying a co-domain to be “the set of positive Real numbers”, you are instructing any who use the function to ignore any negative values it produces.
- The Range can be difficult to specify exactly, but a larger set of numbers that includes the entire Range can be specified. For example, a co-domain could specify the set of all positive Real numbers, even though the function does not generate all possible positive Real numbers.
If the Range is difficult to specify, thinking about a Co-domain can serve the same purpose as thinking about the Range: thinking about which values the function or relation can, and cannot, produce helps one understand the overall behavior of the function or relation.
Why bother with Domain and Range (or Co-Domain)?
While “domain”, “co-domain”, and “range” are indeed “three more vocabulary terms you should know”, they are also useful concepts to master: thinking about them helps you understand the behavior of a function or relation. By determining the domain and range (or co-domain) of a function before you begin to graph it, you will usually develop a mental image of the function that can help you avoid making errors, or wasting space, on your graph paper when graphing the function.
The knowledge that a function “never produces a value less than 5”, as is the case with g(x) above, will affect your choice of values to display along the vertical axis, as well as where you place the origin of your graph. Thinking about the domain and range (or co-domain) of a function before starting to use or graph it will improve your understanding of how it behaves, and should help you work more efficiently.