A polynomial function is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $n$ is a non-negative integer and the coefficients $a_i$ are constants. Finding the domain of a polynomial function is straightforward because polynomial functions are defined for all real numbers. This means you can plug in any real number for $x$, and you'll get a real number output for $f(x)$.
In essence, the domain of any polynomial function is always the set of all real numbers, often written as $(-\infty, \infty)$ or $\mathbb{R}$. This is because there are no restrictions on the values you can input into a polynomial—no division by zero, no square roots of negative numbers, etc. Therefore, practice exercises focus on understanding this fundamental concept and recognizing polynomial forms.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Polynomial Function | A. The set of all possible input values ($x$-values) for which a function is defined. |
| 2. Domain | B. A function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$. |
| 3. Real Numbers | C. The highest power of $x$ in a polynomial function. |
| 4. Degree of a Polynomial | D. All numbers that can be represented on a number line, including rational and irrational numbers. |
| 5. Coefficient | E. The numerical factor of a term in a polynomial. |
Polynomial functions are defined for all ______ _______. This means the _______ of a polynomial function is always $(-\infty, \infty)$. There are no _______ on the values you can input into a polynomial because there is no division by _______ or square roots of _______ numbers.
Explain why the domain of a polynomial function is always all real numbers. Provide an example of a function that is NOT a polynomial and explain why its domain is restricted.
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