Glossary for Lesson 18: Domain and Range

Domain

The domain of a function is the set of all inputs, or values of the independent variable, for which the function has an output.  If some value x is in the domain of a function, then the function is said to be defined at x.  If x is not in the domain, then the function is said to be undefined at x.

Range

The range of a function is the set of all values that the function can take on.  In other words, it is the set of all outputs that the function can produce.

Endpoint

For an interval of any of the four possible types (a,b) , [a,b) , (a,b] , or [a,b] , the numbers a and b are called the endpoints of the interval.  An endpoint, then, is either the upper bound or the lower bound of an interval. On a graph, an endpoint is a point at which a piece of a curve ends.  If the endpoint is represented with an open circle, then the x-value of the endpoint is not included in the domain, and the function is not defined at that point.  If the endpoint is instead a filled-in circle, then the function is defined at that point, and the value of x there is included in the domain.

Increasing Behavior of a Function

We say that a function is increasing on a certain interval if its slope is positive at every point in the region of the graph defined by that interval of the domain.  Put differently, if a and b are two values that lie in an interval in which a function f is increasing, then if a<b , f(a)<f(b).  This means that as we move to the right across an interval where a function is increasing, its graph climbs higher and higher.

Decreasing Behavior of a Function

We say that a function is decreasing on a certain interval if its slope is negative at every point in the region of the graph defined by that interval of the domain.  Put differently, if a and b are two values that lie in an interval in which a function f is decreasing, then if a<b , f(a)>f(b).  This means that as we move to the right across an interval where a function is decreasing, its graph falls lower and lower.

Constant Behavior of a Function

We say that a function is constant on a certain interval if its slope is 0 at every point in the region of the graph defined by that interval of the domain.  Put differently, if a and b are two values that lie in an interval in which a function f is constant, then f(a)=f(b).  This means that on an interval where a function is constant, its graph is just a horizontal line.

Domain Restriction

A domain restriction occurs whenever the domain of a function is not all the real numbers.  In this course, we will talk about two kinds of domain restrictions, those that occur mathematically as a result of a function itself and those that are imposed externally for more practical reasons. As an example of the first type, mathematically imposed domain restrictions, we can look at the function f(x)=\frac {x+2}{x-1} .  Since letting x=1 would make the denominator equal to 0 , and we can’t divide by 0 , 1 is not in the domain.
As an example of the second type of domain restriction, one imposed on a function for non-mathematical reasons, we can think about a real-world example.  Let’s say that we want to look at the how a puppy’s weight, measured in pounds, changes from when it is born until its first birthday.  If this is approximately described by the function w(t)=1+1.5t , where t is the number of weeks that have passed, then we are only interested in values of t in the interval [0,52] , since we’re just considering a limited window of time.  The function w(t)=1+1.5t , taken totally by itself, would have a domain of all the real numbers, but because of the story motivating our use of this function, we will restrict the domain to just [0,52].