Problem 21 - Calculus Early Transcendentals

Question

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. $$p(x)=4 x^{5}-3 x^{2}+1$$

Step-by-Step Solution

Answer

Answer: The function $$p(x)$$ is continuous on the interval $$(-\infty, \infty)$$.

Step 1: p(x) is a polynomial function, as it is a sum of monomials (terms with a single product of a constant and a variable raised to a non-negative integer power). #Step 2: Identify the domain of the function#

Since there are no restrictions on the possible values of x in a polynomial function, the domain of p(x) is all real numbers, which is denoted as $$(-\infty, \infty)$$. #Step 3: Apply Theorem 2.10#

Step 2: Theorem 2.10 states that a polynomial function is continuous at every point on its domain. Since p(x) is a polynomial function, it is continuous on its entire domain. #Step 4: State the intervals of continuity#

The function p(x) is continuous on its entire domain, which is $$(-\infty, \infty)$$.

Key Concepts

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in algebra. A polynomial function, often represented as \( p(x) \), consists of terms called monomials, and each term is made up of a coefficient multiplied by a variable raised to a non-negative integer exponent.

For example, the function \( p(x)=4x^{5}-3x^{2}+1 \) is a polynomial function since it is composed of three monomials: \( 4x^{5} \), \( -3x^{2} \), and the constant term 1. One of the key characteristics of polynomial functions is that they are smooth and continuous curves without breaks, holes, or sharp corners.

Due to these smooth properties, polynomial functions such as quadratic, cubic, and quartic functions exhibit continuity on their entire domain, which is the set of all real numbers, denoted by \( (-ifty, ifty) \). This means that as you graph a polynomial function on a coordinate plane, you can draw it without picking up your pencil, highlighting their continuous nature.

Theorem 2.10 - Continuity of Polynomial Functions

Understanding Theorem 2.10 is crucial when analyzing the continuity of functions. This theorem specifically addresses the continuity of polynomial functions and states that they are continuous at every point within their domain.

The significance of this theorem lies in its assurance that polynomial functions won't have any interruptions or undefined values in their graph. Since the domain of a polynomial function encompasses all real numbers, applying Theorem 2.10 lets us conclude that they are continuous for all \( x \) values, without exception.

This property tremendously simplifies the process of determining the continuity of polynomial functions, as you don't need to search for points of discontinuity meticulously. It's enough to know that as long as the function is a polynomial, its graph will be a seamless curve.

Domain of a Function

The domain of a function can be defined as the complete set of possible values of the independent variable, commonly referred to as \( x \), for which the function is defined.

For different types of functions, the domain might have restrictions depending on their mathematical properties. For instance, for a square root function, the domain only includes numbers for which the radicand is non-negative, since square roots of negative numbers are not real.

In contrast, for polynomial functions like \( p(x)=4x^{5}-3x^{2}+1 \), there are no such restrictions. This is because no matter what real number you plug in for \( x \), you will always get a real number out. Hence, the domain of any polynomial function is all real numbers, \( (-ifty, ifty) \), allowing for an extensive range of inputs and highlighting the function's versatility in various mathematical contexts.