Math Problem Statement
Find each key feature of the function shown in the graph. Enter all intervals using interval notation. Enter the x-intercept as values of x only.
The graph with X-coordinate marks -4, -2, 0, 2, 4 and Y-coordinate mark -4, -2, -1, 0, 2 and 4. There are 3 key features. The first key feature has head at (0, 1) and tail at (-1, 0). The second key feature has head at (1, 0) and tail at (0, -1). The third key feature has head at (2, -1) and tail at (1, -2).
range =
domain =
x-intercept(s) =
average rate of change over the interval [0, 1] =
an interval where the graph is positive =
Solution
The graph shows three segments, and the key features are as follows:
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Range: The range refers to all possible -values that the function can take. From the graph, the -values range from to . Therefore, the range is:
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Domain: The domain refers to all possible -values that the function can take. From the graph, the -values range from to . Therefore, the domain is:
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X-intercepts: The x-intercepts are the points where the graph crosses the -axis. From the graph, the x-intercepts are at and . Therefore, the x-intercepts are:
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Average Rate of Change Over the Interval : The average rate of change is given by the formula: For the interval :
- Therefore, the average rate of change over is:
-
An Interval Where the Graph Is Positive: The graph is positive where the function is above the x-axis. From the graph, the interval where the function is positive is:
If you have any questions or need further explanations, feel free to ask!
Here are 5 related questions to deepen your understanding:
- How do you determine the intervals where a function is increasing or decreasing?
- What is the significance of the slope in determining the rate of change?
- Can a function have more than one x-intercept? What does it mean?
- How would you calculate the average rate of change over a different interval, such as ?
- How do you find the y-intercept from a graph, and what does it represent?
Tip: Remember, the domain and range of a function provide crucial information about the limits of and -values, helping you understand the behavior of the function across its entire graph.
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Math Problem Analysis
Mathematical Concepts
Functions
Graphing
Interval Notation
Average Rate of Change
Formulas
Average Rate of Change formula: (f(b) - f(a)) / (b - a)
Theorems
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Suitable Grade Level
Grades 9-12
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