Functions describe how one quantity depends on another, and their graphs help us see patterns quickly. Some relationships can take every value in an interval, while others only make sense at separate, countable inputs. This difference is called continuous versus discrete, and it affects how we draw, read, and interpret graphs.
Knowing the difference helps prevent mistakes in science, business, statistics, and everyday data analysis.
A continuous function is drawn as an unbroken curve because values exist between any two nearby inputs. A discrete function is shown as separate points because only certain input values are allowed, such as whole numbers of students or days. Continuous graphs often model measurement, motion, temperature, or distance, while discrete graphs often model counts, sequences, and items.
When reading a graph, the key question is whether points between the plotted values are meaningful.
Key Facts
- A continuous function has outputs for every input in an interval, so its graph can be drawn without lifting your pencil.
- A discrete function has outputs only for specific inputs, so its graph is shown as separate points.
- Continuous example: y = 2x + 1 for all real numbers x.
- Discrete example: a(n) = 2n + 1 where n = 0, 1, 2, 3, ...
- Domain means the set of allowed input values, and it determines whether a graph should be continuous or discrete.
- Do not connect discrete points unless values between the points have real meaning.
Vocabulary
- Continuous function
- A function whose inputs can include every value in an interval, producing a graph with no gaps or isolated points.
- Discrete function
- A function whose inputs are separate values, often whole numbers, producing a graph made of individual points.
- Domain
- The set of all input values that are allowed for a function.
- Range
- The set of all output values produced by a function.
- Interpolation
- The process of estimating a value between known data points, which is only appropriate when values between points are meaningful.
Common Mistakes to Avoid
- Connecting every set of plotted points, which is wrong when the data are discrete and values between points do not exist or do not make sense.
- Treating all whole-number inputs as discrete, which is wrong because a function can be defined on real numbers even if only a few sample points are plotted.
- Ignoring the units of the input variable, which is wrong because units such as people, tickets, or cars often require whole-number values.
- Assuming a smooth curve proves the data are continuous, which is wrong because the graph style must match the real meaning of the variables and domain.
Practice Questions
- 1 A movie theater sells tickets for $12 each. Write a function for total cost C in terms of number of tickets n, then state whether the function is continuous or discrete and explain why.
- 2 For the continuous function f(x) = 3x - 2, find f(1.5), f(4), and the change in output from x = 1.5 to x = 4.
- 3 A graph shows the number of students absent each day for 10 school days. Explain whether the points should be connected and what connecting them might incorrectly suggest.