Domain and Range of Polynomial - Algebra-1

1. Fundamental Concepts

Definition: Polynomials are expressions with variables raised to non-negative integer exponents, such as $$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$ where $$a_i \in \mathbb{R}$$ and $$n \geq 0$$ .
Domain: The domain of a polynomial is all real numbers, denoted as $$\mathbb{R}$$ .
Range: The range of a polynomial depends on its degree and leading coefficient. For polynomials of even degree, the range can be all real numbers or a subset of real numbers. For polynomials of odd degree, the range is always all real numbers, $$\mathbb{R}$$ .

Universality of the domain: Regardless of whether the polynomial is of degree 1 (e.g., ), degree 3 (e.g., ), or degree 8 (e.g., ), its domain is always all real numbers, with no exceptions.
Determining factors of the range:
- Parity of the degree:
  - Odd-degree polynomials (degrees 1, 3, 5, etc.): When and , the function values tend to positive infinity and negative infinity (or vice versa), so the range is all real numbers .
  - Even-degree polynomials (degrees 2, 4, 6, etc.): The graph of the function has a minimum point (if the leading coefficient is positive) or a maximum point (if the leading coefficient is negative). Thus, the range is (where k is the function value at the minimum point) or (where k is the function value at the maximum point).
- Sign of the leading coefficient: It only affects the direction of the range of even-degree polynomials (a positive leading coefficient leads to an upward-opening graph with the range lower bounded by the minimum value; a negative leading coefficient leads to a downward-opening graph with the range upper bounded by the maximum value). It has no effect on the direction of the range of odd-degree polynomials.

Problem: Determine the domain and range of the polynomial $$f(x) = x^2 - 4x + 3$$ .

Domain: Since it is a polynomial, the domain is all real numbers, $$\mathbb{R}$$ .
Range: This is a quadratic polynomial (even degree) with a positive leading coefficient. The parabola opens upwards, so the range is $$[y_{\text{min}}, \infty)$$ . To find $$y_{\text{min}}$$ , we complete the square or use the vertex formula. The vertex form is $$f(x) = (x-2)^2 - 1$$ . Thus, the minimum value is $$-1$$ , and the range is $$[-1, \infty)$$ .

Problem: Determine the domain and range of the polynomial $$g(x) = -x^3 + 2x^2 - x + 5$$ .

Domain: Since it is a polynomial, the domain is all real numbers, $$\mathbb{R}$$ .
Range: This is a cubic polynomial (odd degree). The range of any odd-degree polynomial is all real numbers, $$\mathbb{R}$$ .

Example 3: Find the domain and range of (8th-degree, even-degree, positive leading coefficient).

- Domain: All real numbers, .
- Range: The square of is always non-negative (), so the overall function value is . Thus, the range is .

Finding the domain: Directly determine that the domain is all real numbers without additional calculations, as all polynomials are defined for any real number.
Finding the range:
1. Step 1: Determine the parity of the polynomial's degree (odd/even) and the sign of its leading coefficient (positive/negative).
2. Step 2: If it is an odd-degree polynomial, directly conclude that the range is .
3. Step 3: If it is an even-degree polynomial, find the function's maximum or minimum value by completing the square (e.g., for the quadratic function ) or analyzing "non-negative terms + constants" (e.g., for high-degree even polynomials like ), and then write the range as or .