The domain of the absolute value function is all real numbers, and the range is all non-negative real numbers.
Understanding Domain and Range
Let's break down what domain and range mean in the context of the absolute value function, generally expressed as f(x) = |x|.
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Domain: The domain of a function represents all possible input values (x-values) for which the function is defined. In simpler terms, it's the set of all x-values you can plug into the function without causing it to be undefined (like dividing by zero or taking the square root of a negative number).
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Range: The range of a function represents all possible output values (y-values or f(x)-values) that the function can produce. It's the set of all y-values that result from plugging in all the possible x-values from the domain.
The Absolute Value Function
The absolute value function, denoted by |x|, returns the non-negative value of any real number x. In other words:
- If x is positive or zero, |x| = x
- If x is negative, |x| = -x (which makes it positive)
Domain of the Absolute Value Function
You can input any real number into the absolute value function. Whether it's a positive number, a negative number, or zero, the function will always produce a valid output. Therefore, the domain is all real numbers. This can be represented in several ways:
- Set notation: {x | x ∈ ℝ} (x such that x is an element of the set of real numbers)
- Interval notation: (-∞, ∞)
Range of the Absolute Value Function
The output of the absolute value function is always non-negative (zero or positive). It will never return a negative number because it essentially measures the distance from zero. Therefore, the range is all non-negative real numbers. This can be represented as:
- Set notation: {y | y ≥ 0} (y such that y is greater than or equal to 0)
- Interval notation: [0, ∞)
Generalized Absolute Value Functions
For any absolute value function in the form f(x) = |mx + b|, where m and b are real numbers:
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Domain: The domain remains all real numbers (-∞, ∞). You can substitute any real number for x in 'mx + b' without causing a mathematical error.
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Range: The range will always be [0, ∞). The absolute value ensures the output is always non-negative.
