The domain of a function is the set of all possible input values. Depending on the situation represented, a function may take all numbers as its input or only a limited set of numbers.

• Function $A$ gives the area of a square, in square centimeters, as a function of its side length, $s$, in centimeters.

  • The input of $A$ can be 0 or any positive number, such as 4, 7.5, or $\frac{19}{3}$. It cannot include negative numbers because lengths cannot be negative.
  • The domain of $A$ includes 0 and all positive numbers (or $s \geq 0$).

• Function $q$ gives the number of buses needed for a school field trip as a function of the number of people, $n$, going on the trip.

  • The input of $q$ can be 0 or positive whole numbers because a negative or fractional number of people doesn’t make sense.
  • The domain of $q$ includes 0 and all positive whole numbers. If the number of people at a school is 120, then the domain is limited to all non-negative whole numbers up to 120 (or $0 \leq n \leq 120$).

• Function $v$ gives the total number of visitors to a theme park as a function of days, $d$, since a new attraction opened to the public.

  • The input of $v$ can be positive or negative. A positive input means days since the attraction opened, and a negative input means days before the attraction opened.
  • The input can also be whole numbers or fractional. The statement $v(17.5)$ refers to 17.5 days after the attraction opened.
  • The domain of $v$ includes all numbers. If the theme park had opened exactly one year before the new attraction opened, then the domain would be all numbers greater than or equal to -365 (or $d \geq \text-365$).

The range of a function is the set of all possible output values. Once we know the domain of a function, we can determine the range that makes sense in the situation.

• The output of function $A$ is the area of a square in square centimeters, which cannot be negative but can be 0 or greater, not limited to whole numbers. The range of $A$ is 0 and all positive numbers.
• The output of $q$ is the number of buses, which can only be 0 or positive whole numbers. If there are 120 people at the school, however, and if each bus could seat 30 people, then only up to 4 buses are needed. The range that makes sense in this situation would be any whole number that is at least 0 and at most 4.
• The output of function $v$ is the number of visitors, which cannot be fractional or negative. The range of $v$, therefore, includes 0 and all positive whole numbers.