The domain of a polynomial function is always all real numbers, while the range depends on the degree of the polynomial and its leading coefficient.
Understanding Domain and Range
- Domain: The domain of a function represents all possible input values (x-values) for which the function is defined.
- Range: The range of a function represents all possible output values (y-values) that the function can produce.
Domain of Polynomial Functions
For any polynomial function, regardless of its degree or coefficients, you can input any real number and obtain a real number output. Therefore:
- The domain of a polynomial function is all real numbers. This can be represented in interval notation as (-∞, ∞).
Range of Polynomial Functions
The range of a polynomial function is more nuanced and depends on the following factors:
1. Odd-Degree Polynomials
Polynomials with an odd degree (e.g., linear functions, cubic functions) have a range that includes all real numbers. As x approaches positive infinity, y also approaches positive infinity (or negative infinity, depending on the leading coefficient). Similarly, as x approaches negative infinity, y also approaches negative infinity (or positive infinity).
- Therefore, the range of an odd-degree polynomial function is all real numbers, or (-∞, ∞).
2. Even-Degree Polynomials
Polynomials with an even degree (e.g., quadratic functions, quartic functions) have a range that is bounded either above or below. This is because the end behavior of an even-degree polynomial is the same in both directions (either both ends go up or both ends go down).
- Leading Coefficient (a) > 0: If the leading coefficient is positive, the parabola opens upwards. In this case, the range is all real numbers greater than or equal to the minimum value of the function. The minimum value occurs at the vertex of the polynomial. So, the range would be y ≥ k, where k is the y-coordinate of the vertex.
- Leading Coefficient (a) < 0: If the leading coefficient is negative, the parabola opens downwards. In this case, the range is all real numbers less than or equal to the maximum value of the function. The maximum value occurs at the vertex of the polynomial. So, the range would be y ≤ k, where k is the y-coordinate of the vertex.
Example:
Consider the quadratic function f(x) = x2 + 2x + 3.
- The domain is (-∞, ∞).
- Since the leading coefficient (1) is positive, the parabola opens upwards. The vertex can be found using x = -b/2a = -2/(2*1) = -1. Plugging this into the function gives f(-1) = (-1)2 + 2(-1) + 3 = 1 - 2 + 3 = 2. Therefore, the vertex is (-1, 2).
- The range is [2, ∞).
Summary
| Feature | Polynomial Function |
|---|---|
| Domain | All real numbers (-∞, ∞) |
| Range (Odd Degree) | All real numbers (-∞, ∞) |
| Range (Even Degree, a > 0) | y ≥ k (where k is the minimum y-value) |
| Range (Even Degree, a < 0) | y ≤ k (where k is the maximum y-value) |
In conclusion, the domain of any polynomial function is always all real numbers. However, the range depends on whether the polynomial's degree is odd or even and, in the case of even-degree polynomials, on the sign of the leading coefficient and the location of the vertex.
