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Chapter 12: Problem 60

What is the domain of a polynomial function?

Short Answer

Expert verified
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 by step solution

01

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: - \(f(x) = 5x^3 - 2x^2 + 6\) - \(g(x) = x^4 - 3x + 7\)
02

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.
03

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.
04

Final answer

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

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Most popular questions from this chapter

Determine the domain of each function. For labor only, a plumber charges \(\$ 30\) for a repair visit plus \(\$ 60\) per hour. These labor charges can be described by the function \(L(h)=60 h+30,\) where \(h\) is the time, in hours, and \(L\) is the cost of labor, in dollars. A. Find \(L(2)\) and explain what this means in the context of the problem. B. Find \(L(1)\) and explain what this means in the context of the problem. C. Find \(h\) so that \(L(h)=210,\) and explain what this means in the context of the problem.

Let \(f(x)=[x] .\) Find the following function values. $$f(9.2)$$

Graph the following greatest integer functions. $$g(x)=[x+2]$$

A company's revenue, \(R(x)\) in dollars, from the sale of \(x\) dog houses is given by \(R(x)=60 x\). The company's cost, \(C(x)\) in dollars, to produce \(x\) dog houses is \(C(x)=45 x+6000\). a) Find the profit function, \(P(x),\) that describes the company's profit from the sale of \(x\) dog houses. b) What is the profit from the sale of 300 dog houses?

Oil spilled from a ship off the coast of Alaska with the oil spreading out in a circle across the surface of the water. The radius of the oil spill is given by \(r(t)=4 t\) where \(t\) is the number of minutes after the leak began and \(r(t)\) is in feet. The area of the spill is given by \(A(r)=\pi r^{2}\) where \(r\) represents the radius of the oil slick. Find each of the following and explain their meanings. (IMAGE CANT COPY) a) \(r(5)\) b) \(\quad A(20)\) c) \(A(r(t))\) d) \(A(r(5))\)
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