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

Verified step by step guidance

1. Identify the given function and write it down explicitly to understand its form and any potential points of discontinuity.
2. Recall that polynomial functions are continuous everywhere, rational functions are continuous where the denominator is not zero, and root functions are continuous where the expression inside the root is defined (e.g., non-negative for even roots).
3. Find the domain of the function by determining where it is defined. For rational functions, set the denominator not equal to zero and solve for the variable. For root functions, set the radicand greater than or equal to zero and solve.
4. Use the domain to identify intervals where the function is continuous. The function is continuous on all intervals within its domain where no discontinuities (like division by zero or negative radicands) occur.
5. Express the intervals of continuity using interval notation, combining all continuous intervals found from the domain analysis.

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 of the function as it approaches that point equals the function's value there. Continuity over an interval means the function has no breaks, jumps, or holes within that interval. Recognizing types of discontinuities helps in determining continuous intervals.

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 pinpointing where continuity fails and thus in determining intervals where the function remains continuous.