Determine the intervals of the domain over which each function is continuous. See Example 1.

Verified step by step guidance

1. Identify the given function and write it down explicitly to understand its form and components.
2. Recall that polynomial, rational, root, logarithmic, and trigonometric functions have different continuity properties. For example, polynomials are continuous everywhere, while rational functions may be discontinuous where the denominator is zero.
3. Determine the domain of the function by finding all values of \(x\) for which the function is defined. This often involves setting denominators not equal to zero, ensuring expressions under even roots are non-negative, and arguments of logarithms are positive.
4. Once the domain is established, analyze the function's behavior on that domain. Since most elementary functions are continuous on their domains, the function will be continuous on intervals where it is defined without breaks or holes.
5. Express the intervals of continuity by writing the domain intervals where the function is continuous, using interval notation such as \((-\infty, a)\), \((a, b)\), or \([c, \infty)\), depending on the domain restrictions found.

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. Understanding the domain is essential because continuity can only be analyzed where the function exists. For example, functions involving square roots or denominators require restrictions to avoid undefined values.

Continuity of a Function

A function is continuous at a point if the limit as x approaches that point equals the function's value there. Continuity over an interval means the function has no breaks, jumps, or holes in that interval. Recognizing types of discontinuities helps determine where the function is continuous.

Types of Discontinuities

Discontinuities occur when a function is not continuous at a point. Common types include removable (holes), jump, and infinite discontinuities. Identifying these helps in determining intervals where the function remains continuous by excluding points of discontinuity.