Understanding Functions: Domain, Range, and Equations

Functions and Their Graphs

Functions can be either continuous or discrete. Continuous functions have no breaks between input and output values - they include all possible values including decimals and can be drawn as unbroken lines. Discrete functions, on the other hand, are represented by separate dots on a graph since they only include distinct, separated values.

When working with functions, you need to understand domain and range. The domain is the set of all x-values (inputs) in the function, while the range is the set of all y-values (outputs). For example, in a function with coordinates (2,3) and (4,9), the domain would be {2,4} and the range would be {3,9}.

To find the domain and range from a graph, look for the farthest x-values (domain) and the farthest y-values (range). For continuous functions, the domain might include all real numbers, written as D: all x where x is a real number. The range might be restricted, such as R: y≥3, meaning all y-values that are 3 or higher.

Remember This! When evaluating functions like f$x$ = x + 2, the entire f$x$ represents the y-value. So if you're asked to find f$-1$, you substitute -1 wherever you see x in the equation: f$-1$ = -1 + 2 = 1.