The domain and range of a power function depend on the specific form of the function, particularly the exponent. Let's explore this in more detail.
A power function is generally defined as f(x) = kxp, where k and p are constants. The domain and range are influenced by whether p is an integer (positive, negative, or zero) or a fraction, and whether p is even or odd.
Understanding Domain and Range
Before diving into specifics, let's define these terms:
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Domain: The set of all possible input values (x-values) for which the function is defined. According to the provided reference, "The domain of a function is the set of input values that are used for the independent variable."
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Range: The set of all possible output values (y-values) that the function can produce. According to the provided reference, "The range of a function is the set of output values for the dependent variable".
Power Function Cases and Their Domains & Ranges
Here’s a breakdown of the domain and range for different types of power functions:
| Case | Function Example | Domain | Range |
|---|---|---|---|
| Positive Integer Exponent (p > 0) | f(x) = x2 | All real numbers (-∞, ∞) | [0, ∞) if p is even; (-∞, ∞) if p is odd |
| Negative Integer Exponent (p < 0) | f(x) = x-1 = 1/x | All real numbers except 0 (-∞, 0) U (0, ∞) | All real numbers except 0 (-∞, 0) U (0, ∞) |
| Zero Exponent (p = 0) | f(x) = x0 = 1 | All real numbers except possibly 0 | {1} |
| Fractional Exponent (p = 1/n) | f(x) = x1/2 = √x | [0, ∞) | [0, ∞) |
Examples and Practical Insights:
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f(x) = x2 (Parabola): This is a classic power function. You can input any real number (positive, negative, or zero). However, the output is always non-negative because squaring a number always results in a positive value or zero. Thus, the domain is all real numbers, and the range is [0, ∞).
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f(x) = x3 (Cubic Function): Again, you can cube any real number. Cubing positive numbers results in positive numbers, cubing negative numbers results in negative numbers, and cubing zero results in zero. So, the domain and range are both all real numbers (-∞, ∞).
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f(x) = 1/x (Reciprocal Function): You can input any real number except 0 because division by zero is undefined. As x approaches 0, the function approaches infinity (positive or negative, depending on the side). The function can take on any real value except 0. Therefore, both the domain and the range are all real numbers except 0.
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f(x) = √x (Square Root Function): You can only take the square root of non-negative numbers (in the real number system). The result of a square root is always non-negative. Therefore, both the domain and the range are [0, ∞).
The reference states that for an exponential function f(x) = abx, the domain is all real numbers. While related, exponential functions are different from power functions; in exponential functions, the variable is in the exponent, whereas in power functions, the variable is the base.
