https://file.aiquickdraw.com/mathbot/file/1726192999353ewrbm4ld.jpg

Explain the answer in detail and solve the problem step by step eith proof 

Let's analyze and solve the problem step by step.

### Step 1: Understanding the graph and the function
In the image, the function shown is a straight line starting from the point \((-6, -10)\) and increasing steadily up to the point \((8, 10)\).

A straight line typically represents a linear function of the form \( y = mx + b \), where:
- \( m \) is the slope (rate of change of \( y \) with respect to \( x \)),
- \( b \) is the y-intercept (the point where the line crosses the y-axis).

From the graph, it appears that the line is continuous (no breaks, jumps, or holes) over the entire visible domain.

### Step 2: Identifying the domain
The domain of the function is the set of all possible \( x \)-values for which the function is defined. From the graph, the \( x \)-values span from \( -6 \) to \( 8 \), as those are the endpoints where the line is drawn.

Thus, the domain of this function is:
\[
[-6, 8]
\]

### Step 3: Continuity of the function
A function is continuous over an interval if there are no breaks, jumps, or asymptotes within the interval. Since the given function is a straight line with no interruptions or breaks, it is continuous over the entire domain.

### Step 4: Conclusion
The function is continuous over the interval \([-6, 8]\).

### Step 5: Answering the question
From the provided multiple-choice options:

- **Option A** states that the function is continuous over an interval. This is correct.
- The specific interval is \([-6, 8]\), which you would type in interval notation as the answer.

Thus, the correct choice is:
\[
\boxed{A. \text{The function is continuous over the interval } [-6, 8].}
\]

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Do you have any more questions or would you like further details on any part of this solution?

### Here are 5 related questions to extend your understanding:
1. What are the conditions for a function to be continuous over a given interval?
2. How do you determine the slope of a linear function from a graph?
3. What would happen to the continuity if there was a hole or a jump in the graph?
4. How would you express the range of the given function based on the graph?
5. Can a function have a discontinuity at only one point and still be continuous elsewhere?

### Tip:
When determining continuity from a graph, always check for visible breaks, jumps, or holes in the graph. A smooth, unbroken line or curve generally indicates continuity over the visible domain.

Determine the intervals of the domain over which the function shown in the graph is continuous.

Solve a problem involving the continuity of a linear function from the graph provided. Learn how to determine the interval where the function is continuous, covering key algebraic concepts such as linear functions, continuity, and interval notation. Suitable for high school students.

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