Problem 66 - Brief Applied Calculus

Question

Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ f(x)=5 x^{3}-6 x^{2}+2 x-4 $$

Step-by-Step Solution

Answer

The function is continuous everywhere on the real number line; it has no points of discontinuity.

Step 1: Understand the Function Type

The given function is a polynomial function. Polynomial functions are continuous for all real values of x. This is a fundamental property of polynomials as they are made up of sums, differences, and products of continuous functions (in this case, monomials of non-negative integer powers of x).

Step 2: Evaluate Continuity

Based on the property identified in Step 1, understand that continuous functions have no gaps, jumps, holes, or points of discontinuity across their domains. Since polynomial functions are entire functions, they are continuous everywhere on the real number line, which includes all possible values of x.

Key Concepts

Polynomial Functions

Polynomial functions are a very important class of functions in mathematics. They are expressions involving variables raised to whole number exponents and multiplied by coefficients. For example, the function \( f(x) = 5x^{3} - 6x^{2} + 2x - 4 \) is a polynomial of degree 3.

• The degree of a polynomial is determined by the highest exponent in the function, which in this case is 3 for \( x^3 \).
• Each term consists of a coefficient and a variable raised to an integer power, such as 5 in \( 5x^{3} \).
• Polynomials can have multiple terms with different degrees, and they can be as simple or as complex as needed.

Polynomials are algebraic expressions used in a variety of applications, including engineering and physics, thanks to their versatile nature.

Continuous Functions

Continuity is a key concept in calculus and pre-calculus, and it refers to how smoothly functions behave. A function is continuous at a specific point if there is no interruption in the graph of the function at that point.
For a function to be continuous:

• The limit of the function as it approaches a specific point from the left must equal the limit as it approaches that point from the right.
• The value of the function at that specific point must equal the limit at that point.
• This is the general idea: no sudden jumps, gaps, or breaks in the function's graph.

This smooth behavior means continuity is crucial for analyzing real-world scenarios that require stable results. Polynomial functions are known for being continuous; since they don't have breaks or abrupt changes, they maintain this smooth behavior over their entire domain.

Real Number Line

The real number line is an essential concept in mathematics, representing all possible real numbers in a sequential manner. These real numbers include all the integers, fractions, and irrational numbers that exist.

• The real number line allows easy visualization of how numbers are ordered and spaced relative to one another.
• On the real number line, every point corresponds to a unique real number, and each real number corresponds to a distinct point.
• This enables mathematicians and students alike to see relationships between numbers, especially when working with functions.

Due to their continuous nature, polynomial functions stretch seamlessly across the real number line, meaning they are defined and smooth everywhere without exception. This characteristic reflects why they are such important tools for graphing and solving equations in both theoretical and applied settings.