Understanding domain and range is fundamental to grasping functions in mathematics. This guide provides a clear explanation of these concepts, illustrated with examples, and suitable for note-taking. We'll explore various function types and how to determine their domain and range.
What is the Domain of a Function?
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the x-values you can plug into the function and get a real, defined y-value back. The domain is often restricted by certain mathematical rules, which we'll explore below.
Identifying the Domain: Common Restrictions
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Division by Zero: A function is undefined when the denominator is zero. You must exclude any x-values that would lead to division by zero.
- Example: For the function f(x) = 1/(x-2), the domain is all real numbers except x = 2. We write this as: Domain: (-∞, 2) U (2, ∞)
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Even Roots of Negative Numbers: The square root (and other even roots) of a negative number is not a real number. Therefore, the expression under the even root must be greater than or equal to zero.
- Example: For the function g(x) = √(x+3), the expression under the square root (x+3) must be non-negative. Solving x + 3 ≥ 0 gives x ≥ -3. Thus, the domain is: Domain: [-3, ∞)
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Logarithms: The argument of a logarithm must be positive.
- Example: For the function h(x) = log₂(x-1), x - 1 must be greater than zero. This means x > 1. Domain: (1, ∞)
What is the Range of a Function?
The range of a function is the set of all possible output values (y-values) that the function can produce. It represents the complete set of values the function can attain.
Determining the Range: Methods and Examples
Finding the range can be more challenging than finding the domain. Several methods exist, depending on the function's type:
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Graphical Method: The range is easily visualized by examining the graph of the function. The range encompasses all y-values where the graph exists.
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Algebraic Method: This involves solving for x in terms of y and then identifying the restrictions on y. This is particularly useful for simpler functions.
- Example: Let's reconsider f(x) = 1/(x-2). To find the range, we solve for x: y(x-2) = 1 => xy - 2y = 1 => xy = 1 + 2y => x = (1+2y)/y. Notice that y cannot be zero. Therefore, the range is all real numbers except y = 0. Range: (-∞, 0) U (0, ∞)
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Using the Domain and Function Behavior: Understanding the domain can often shed light on the range. Analyzing the function's behavior (increasing, decreasing, asymptotes, etc.) can further help in determining the range.
Examples of Finding Domain and Range
Let's analyze a few more examples:
1. f(x) = x² + 2
- Domain: All real numbers (-∞, ∞) because you can square any real number.
- Range: [2, ∞) because x² is always non-negative, so x² + 2 is always greater than or equal to 2.
2. g(x) = √(4 - x²)
- Domain: The expression inside the square root must be non-negative: 4 - x² ≥ 0 => x² ≤ 4 => -2 ≤ x ≤ 2. Domain: [-2, 2]
- Range: [0, 2] The square root is always non-negative, and the maximum value occurs at x=0, resulting in √4 = 2.
3. h(x) = |x| -1
- Domain: All real numbers (-∞, ∞) as the absolute value is defined for all real numbers.
- Range: [-1, ∞) The absolute value is always non-negative, therefore |x| - 1 will always be greater than or equal to -1.
This guide provides a foundational understanding of domain and range. Remember to always consider the specific restrictions imposed by the function's type when determining its domain and range. Practice with various functions to solidify your understanding.
