Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=x^{2}-2 x+1\)

Step-by-Step Solution

Answer

The function \(f(x) = {x^{2} - 2x + 1}\) is continuous on the interval \(-\infty < x < \infty\), as it is a polynomial function which is continuous for all real numbers. It has no discontinuities.

Step 1: Determine the type of function

In this case, the function \(f(x) = {x^{2} - 2x + 1}\) is a polynomial function. It is important to recognize the type of function to apply the correct rules of continuity.

Step 2: Apply polynomial function rule

Polynomial functions are continuous everywhere. That means that their graph can be traced without lifting the pen. Hence, the function \(f(x) = {x^{2} - 2x + 1}\) is continuous on all real numbers, or we can write this interval as \(-\infty < x < \infty\).

Step 3: Evaluate the conditions of continuity

For a function to be continuous at a point, the function must satisfy the following three conditions: \(f(a)\) is defined, \(\lim_{x \to a} f(x)\) exists, and \(\lim_{x \to a} f(x) = f(a)\). Since a polynomial function satisfies all these three conditions for any real number, there are therefore no discontinuities in the function.

Key Concepts

Polynomial Functions

Polynomial functions are among the most fundamental concepts in mathematics. A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In its simplest form, it has the structure \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). Here, each term consists of a coefficient (like \(a_n\)) and a variable raised to a whole number power. Polynomial functions are smooth and unbroken, which is a key characteristic leading to their continuity. When dealing with polynomials like \(f(x) = x^2 - 2x + 1\), they are continuous across the entire set of real numbers. This means you can trace their graph in one motion without any breaks.

In practical terms, understanding polynomial functions involves:

• Identifying the degree of the function (highest power of \(x\)).
• Determining the coefficients and constants.
• Understanding the general shape of the graph (quadratic, cubic, etc.).

Recognizing the behavior of polynomial functions helps in solving many mathematical problems and situations, where continuity plays a crucial role.

Conditions of Continuity

Continuity is essential in understanding how smooth or uninterrupted a function is. For a function to be continuous at a given point, it must satisfy three important conditions:

• The function value \(f(a)\) must be defined.
• The limit of the function as \(x\) approaches \(a\), denoted as \(\lim_{x \to a} f(x)\), must exist.
• The value of the limit of the function at \(a\) must equal the function value \((\lim_{x \to a} f(x) = f(a))\).

These conditions ensure that there are no jumps, holes, or vertical asymptotes at the point in question. Since polynomial functions, by their nature, do not break any of these rules for any real number \(x\), they are inherently continuous.

For our specific polynomial, \(x^2 - 2x + 1\), these conditions hold true universally, meaning the function is continuous without exception. Recognizing these conditions helps in assessing whether any function might have discontinuities, an essential skill in both calculus and real analysis.

Interval of Continuity

Identifying the interval of continuity of a function means finding all the input values \(x\) for which the function is continuous. For polynomial functions, such as \(f(x) = x^2 - 2x + 1\), we know that they are continuous over all real numbers. This means that their interval of continuity is \((-\infty, \infty)\).

When dealing with other types of functions, especially rational, trigonometric, or piecewise functions, the interval of continuity can be more limited due to the nature of their potential discontinuities. But for polynomial functions, you can confidently state that there are no breaks, holes, or jumps regardless of the value of \(x\).

• Polynomials have an entire real line as their interval of continuity.
• They are one of the most straightforward functions to evaluate for continuity.
• Understanding polynomial intervals is pivotal in higher-level mathematics courses.

Thus, when asked to find the interval of continuity for a polynomial, it is often as simple as stating that it is continuous everywhere. This makes polynomial functions both versatile and reliable for numerous mathematical applications.