Problem 43 - Calculus Early Transcendentals

Question

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. $$f(x)=\sqrt[3]{x^{2}-2 x-3}$$

Step-by-Step Solution

Answer

Answer: The function is continuous for all real numbers, i.e., the interval is $$(-\infty, \infty)$$.

Step 1: Identify the function and expression inside the cube root

We are given the function $$f(x) = \sqrt[3]{x^2 - 2x - 3}$$ and the expression inside the cube root is $$x^2 - 2x - 3$$.

Step 2: Analyzing continuity of the expression inside the cube root

As a quadratic polynomial, the expression $$x^2 - 2x - 3$$ is continuous on the entire real number line. This means there are no values of x that would make the expression undefined.

Step 3: Determining continuity of the given function

The cube root function is continuous for all values in its domain. Since the expression inside the cube root is continuous for all real numbers (as found in step 2), it means that the given function $$f(x) = \sqrt[3]{x^2 - 2x - 3}$$ is also continuous for all real numbers.

Step 4: Write the continuous intervals

Since the function is continuous for all real numbers, the interval on which it is continuous is: $$(-\infty, \infty)$$

Key Concepts

Intervals of Continuity

In mathematics, determining the intervals of continuity for a function helps us understand where the function behaves in a smooth and predictable manner. A function is said to be continuous on an interval if there are no breaks, jumps, or holes in its graph along that interval.
To determine this, we look for any values of the variable that would disrupt the graph of the function, such as division by zero, taking the square root of a negative number (for real numbers), or other operations that could lead to undefined values.

• For right-continuity, we consider how the function behaves as it approaches from the left to a specific point.
• For left-continuity, we analyze the function's behavior as it reaches that point from the right.
• If the function approaches the same value from both directions, it is continuous at that point.

In our case, the function is continuous on the interval \( (-\infty, \infty) \), meaning it smoothly fits every real number without interruption.

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients. They involve operations of addition, subtraction, multiplication, and non-negative integer exponents on the variables.
These expressions are a cornerstone of algebra due to their simple structure and versatile applications. A polynomial of degree 2, for example, is called a quadratic polynomial and has a general form of \( ax^2 + bx + c \).

• Polynomials are continuous everywhere on the real number line, meaning they pose no discontinuities like breaks or gaps.
• The graph of a polynomial function is smooth and unbroken.
• Polynomials can have factors, which might suggest points of potential discontinuity, but these are typically removable by simplifying.

In the given problem, the polynomial part, \( x^2 - 2x - 3 \), clearly displays this continuity, ensuring smooth operation of the overall function.

Cube Root Function

Cube root functions involve finding a number which, when multiplied by itself three times, gives the original number. The standard cube root function is denoted as \( \sqrt[3]{x} \), and unlike square roots, cube roots are defined for all real numbers, including negatives.

• This property of cube roots makes them inherently continuous across the entire real number set.
• No restrictions like non-negatives apply, as they do with square roots.
• The cube root function can be represented graphically with a curve that smoothly extends in both directions of the real number axis.

So, when we have a cube root function nested over a continuous polynomial, like in the exercise, \( f(x) = \sqrt[3]{x^2 - 2x - 3} \),we further ensure the function remains continuous across all real numbers.