Problem 15 - Calculus for Biology and Medicine

Question

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=3 x^{4}-x^{2}+4 $$

Step-by-Step Solution

Answer

The function \(f(x) = 3x^4 - x^2 + 4\) is continuous for all real numbers \(x \in \mathbf{R}\).

Step 1: Understanding Continuity of Polynomials

A function is continuous at a point if there is no interruption at that point in its graph. For polynomials, which are continuous everywhere on the real number line, we examine if this function is polynomial.

Step 2: Identify the Given Function

The given function is \(f(x) = 3x^4 - x^2 + 4\). This is a polynomial because it is composed of terms that are constant multiplied by a variable raised to a whole number power.

Step 3: Continuous Nature of Polynomials

Since any polynomial function is inherently continuous for all real numbers, \(f(x)\) is continuous wherever it is defined, which means it is continuous for all \(x \in \mathbf{R}\).

Step 4: Conclusion - Application to Problem

For this specific polynomial function, since there are no discontinuities like division by zero or undefined expressions, it maintains continuity over its entire domain.

Key Concepts

Understanding Polynomials

Polynomials are fascinating mathematical expressions consisting of variables and coefficients, where the variable is often raised to a whole number power. The structure of a polynomial expression is a sum of such terms. For instance, a simple polynomial could be

• \( ax^n + bx^{n-1} + \, ... \, + k \), where each letter represents a coefficient and \( n \) is a non-negative integer.

This structure is straightforward, yet powerful, as it allows for a wide array of manipulations and simplifications.
Polynomials can take many forms, and they are characterized by their degree, which is determined by the highest power of the variable present. For example, a polynomial like \( 3x^4 - x^2 + 4 \) is a fourth-degree polynomial due to the term \( 3x^4 \). Understanding these basic components can demystify their seemingly complex nature.
Where polynomials truly shine is in their simplicity and smoothness, making them quick to identify within expressions and equations.

Exploring Continuous Functions

A function is considered continuous if there is no abrupt change or interruption in its graph. In simple terms, you can draw the graph without lifting your pencil. For polynomials, this continuous property is always naturally upheld.
Why are polynomials always continuous? It's because they don't have any restrictions like division by zero or square roots of negative numbers that could disrupt the graph. They are smooth and unbroken across their domain. This intrinsic nature makes them an essential study in calculus and analysis.
To determine if a function is continuous, we commonly check if it fits simple criteria such as limits existing at every point and matching function values. For polynomial functions, this is automatically satisfied over all real numbers, making them ideal examples of continuous functions.

Role of Real Numbers in Function Continuity

The realm of real numbers, denoted as \( \mathbf{R} \), is vast, covering every number on the number line that you can think of—whole numbers, fractions, and irrational numbers included. Because it encompasses such a broad range, polynomials can confidently claim continuity over every real number.

• This means you can input any real number into the polynomial function without causing any disruption in its behavior.
• Each real number results in a smooth output of the function, seamlessly connecting to the next.

This unbroken flow of values makes studying polynomial functions straightforward and reliable. Polynomial functions like \( f(x) = 3x^4 - x^2 + 4 \) do not have any restrictions when it comes to real numbers, thus maintaining their continuity for all \( x \in \mathbf{R} \). Understanding this connection between real numbers and function continuity is crucial for grasping more advanced mathematical concepts.