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

Verified step by step guidance

1. Step 1: Identify the domain of the function from the graph. The graph shows a curve starting from the left and approaching the point (5, 0), which is included as a solid dot, indicating the function is defined at x = 5.
2. Step 2: Understand that a function is continuous on an interval if there are no breaks, jumps, or holes in the graph over that interval. The graph appears smooth and connected from the left up to x = 5.
3. Step 3: Check the behavior at x = 5. Since the graph has a solid dot at (5, 0) and the curve approaches this point from the left, the function is continuous at x = 5.
4. Step 4: Notice that the graph does not extend beyond x = 5 to the right, so the domain ends at x = 5. Therefore, the function is continuous on the interval from negative infinity to 5, inclusive of 5.
5. Step 5: Write the interval of continuity as \((-\infty, 5]\), indicating the function is continuous for all x less than or equal to 5.

Key Concepts

Continuity of a Function

A function is continuous at a point if the limit of the function as it approaches the point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph at that point. Continuity over an interval means the function is continuous at every point within that interval.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Understanding the domain is essential to determine where the function exists and to analyze continuity only within those intervals where the function is defined.

Interpreting Graphs for Continuity

Graphs visually show where a function is continuous or discontinuous. Points where the graph is unbroken and connected indicate continuity, while gaps, jumps, or isolated points indicate discontinuities. The filled point at (5, 0) suggests the function is defined and continuous at x=5.