Problem 34 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

List all values of \(x\) for which the given function is not continuous. $$ f(x)=5 x^{3}-3 x+\sqrt{x} $$

Step-by-Step Solution

Answer

The function \(f(x)\) is not continuous for \(x < 0\).

Step 1: Identify the components of the function

Consider the function given: \( f(x) = 5x^3 - 3x + \sqrt{x} \). It consists of three terms: \( 5x^3 \), \( -3x \), and \( \sqrt{x} \).

Step 2: Analyze the polynomial part

Both \( 5x^3 \) and \( -3x \) are polynomial terms. Polynomial functions are continuous for all real numbers. This means the terms \( 5x^3 \) and \( -3x \) are continuous everywhere.

Step 3: Analyze the square root term

Consider the term \( \sqrt{x} \). The square root function \( \sqrt{x} \) is continuous for all \( x \geq 0 \) but is not defined for \( x < 0 \).

Step 4: Determine the points of discontinuity

Since \( f(x) = 5x^3 - 3x + \sqrt{x} \) consists of a polynomial part (continuous everywhere) and a square root part (continuous for \( x \geq 0 \) but undefined for \( x < 0 \)), the function \( f(x) \) is not continuous for \( x < 0 \).

Key Concepts

polynomial functions

Polynomial functions are a key concept in the study of continuity in functions. They are expressions that involve variables raised to whole number powers, and can be written as combinations of terms like: \(a_nx^n + a_{n-1}x^{n-1} + \, \text{...} \, + a_1x + a_0\), where \(a_n, a_{n-1}, \, \text{...} \, , a_1,\) and \(a_0\) are coefficients.
What makes polynomial functions special is their continuity property. Polynomial functions are continuous everywhere. This means there are no breaks, jumps, or holes in their graph.

For our specific function, \(f(x) = 5x^3 - 3x + \sqrt{x}\), we observe that the terms \(5x^3\) and \(-3x\) , represent polynomial parts. Therefore, both these terms are continuous for all real numbers.

square root function

The square root function is another important concept to understand when discussing continuity. A square root function is written as \(\sqrt{x}\). This type of function is only defined for non-negative values of \(x\) (i.e., \(x \geq 0\)).
This is because the square root of a negative number does not result in a real number.
In our function, the term \(\sqrt{x}\) limits the domain of \(f(x)\). Specifically, it means that \(f(x)\) is only defined for \(x \geq 0\). If \(x < 0\), \(\sqrt{x}\) becomes undefined, making the whole function discontinuous.

So, while the polynomial terms \(5x^3\) and \(-3x\) are continuous everywhere, it is the square root term that introduces a restriction.

points of discontinuity

Points of discontinuity are values of \(x\) where a function is not continuous. A function can be continuous over its domain, but there might be specific points where it fails to be continuous. To find points of discontinuity, we analyze the function and determine where it might be undefined or behaves improperly.
For \(f(x) = 5x^3 - 3x + \sqrt{x}\), we need to consider the individual terms.
- For the polynomial parts \(5x^3\) and \(-3x\), there are no points of discontinuity since they are continuous everywhere.
- For the square root part \(\sqrt{x}\), the term is only defined for \(x \geq 0\).

This means our function \(f(x)\) will be discontinuous or not defined for any \(x < 0\). So, the points of discontinuity are all negative numbers \(x < 0\).