The domain and range of a typical linear equation are all real numbers.
Understanding Domain and Range
- Domain: The domain represents all possible input values (x-values) that a function can accept. According to the reference, the domain is "the set of all the x-coordinates on the function's graph."
- Range: The range represents all possible output values (y-values) that a function can produce. According to the reference, the range is "the set of all y-coordinates" on the function's graph.
Domain and Range of Linear Equations
Most linear equations, when graphed, produce a straight line that extends infinitely in both directions. This means:
- The line covers all possible x-values.
- The line covers all possible y-values.
Therefore, the domain and range are all real numbers, which can be expressed in several ways:
- Interval Notation: (-∞, ∞)
- Set Notation: {x | x ∈ ℝ} (meaning "all x such that x is a real number")
Exceptions
There are a couple of exceptions to the "all real numbers" rule:
-
Horizontal Lines: A horizontal line has the equation y = c, where 'c' is a constant. In this case:
- The domain is still all real numbers (-∞, ∞).
- The range is just the single value 'c': {c}.
-
Vertical Lines: A vertical line has the equation x = c, where 'c' is a constant. In this case:
- The domain is just the single value 'c': {c}.
- The range is all real numbers (-∞, ∞). However, it's important to note that vertical lines technically are not functions because they fail the vertical line test.
Examples
| Equation | Domain | Range |
|---|---|---|
| y = 2x + 1 | (-∞, ∞) | (-∞, ∞) |
| y = 5 | (-∞, ∞) | {5} |
| x = -3 (Not a function) | {-3} | (-∞, ∞) |
| y = -x + 7 | (-∞, ∞) | (-∞, ∞) |
