Problem 60 - Beginning and Intermediate Algebra

Question

What is the domain of a polynomial function?

Step-by-Step Solution

Answer

The domain of a polynomial function is the set of all real numbers, which can be written in interval notation as: Domain: \((-\infty, +\infty)\)

Step 1: Understanding polynomial functions

Polynomial functions are mathematical expressions involving a sum of terms, each term being a product of a constant and a variable raised to a non-negative integer exponent. A polynomial function can be written as: \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constant coefficients, \(n\) is a non-negative integer, and \(x\) is a variable. Examples of some polynomial functions are: - \(f(x) = 5x^3 - 2x^2 + 6\) - \(g(x) = x^4 - 3x + 7\)

Step 2: Defining the domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of x-values that can be used without creating any undefined results such as division by zero or complex numbers (when dealing with real numbers). In the case of a polynomial function, we need to identify if there are any limitations to the input values that we can use.

Step 3: Finding the domain of a polynomial function

Since the exponents in a polynomial function are always non-negative integers and there is no division involved, there is no restriction on the x-values we can use. Therefore, a polynomial function will always be defined for all real numbers.

Step 4: Final answer

The domain of a polynomial function is the set of all real numbers, which can be written in interval notation as: Domain: \((-\infty, +\infty)\)