Problem 45 - Calculus: Early Transcendentals

Question

Functions with roots Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is \(f\) continuous from the left or continuous from the right? $$f(x)=\sqrt[3]{x^{2}-2 x-3}$$

Step-by-Step Solution

Answer

Based on the step by step solution, answer the following question: Question: Determine the interval(s) on which \(f(x) = \sqrt[3]{x^2 - 2x - 3}\) is continuous and whether the function is continuous from the left or continuous from the right at the finite endpoints of the intervals. Answer: The function \(f(x) = \sqrt[3]{x^2 - 2x - 3}\) is continuous on the interval \((-\infty, \infty)\), and since there are no finite endpoints, the function is continuous from both the left and right at every point on the real number line.

Step 1: Find the domain of the function inside the cube root

For a cube root function, we need to find the domain of the expression inside the cube root. In this case, we have a quadratic function: $$x^2 - 2x - 3$$ Since this is a quadratic function, it is defined for all real numbers. So, the domain of \(f(x)\) is all real numbers. There is no restriction on the continuity of the function.

Step 2: Identify the intervals of continuity

Since the function is continuous for all real numbers, we can say that the intervals of continuity are \((-\infty, \infty)\).

Step 3: Analyzing Continuity from the Left and Right at Finite Endpoints

Since the function is continuous for all real numbers and has no finite end points of discontinuity, the function is continuous from both the left and right sides at every point on the real number line.

Key Concepts

Domain of a Function

Understanding the domain of a function is fundamental for studying its continuity. The domain of a function consists of all the possible input values (usually