# 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].