The domain and range of a standard linear function are both the set of all real numbers.
Understanding Domain and Range
- Domain: The domain of a function represents all possible input values (typically x-values) for which the function is defined. In other words, what values can you plug into the function?
- Range: The range of a function represents all possible output values (typically y-values) that the function can produce. In other words, what values come out of the function?
Linear Functions and Real Numbers
A linear function is generally expressed in the form f(x) = mx + b, where m and b are constants.
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Domain (All Real Numbers): You can substitute any real number for x in the equation f(x) = mx + b, and you'll always get a valid result. There are no restrictions. Therefore, the domain is all real numbers, which can be written as (-∞, ∞).
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Range (All Real Numbers): Similarly, for any real number y, you can always find an x such that f(x) = y. A linear function extends infinitely in both positive and negative y-directions. Therefore, the range is also all real numbers, which can be written as (-∞, ∞).
Exceptions: Restricted Linear Functions
It's important to note that while most linear functions have a domain and range of all real numbers, there can be exceptions if the function is explicitly restricted.
For example:
- If we define a linear function only for a specific interval, like f(x) = 2x + 1 for 0 ≤ x ≤ 5, then the domain is [0, 5] and the range would be [1, 11].
However, without specific restrictions, assume the standard linear function encompasses all real numbers for both the domain and range.
