Math Problem Statement
Domain of function: Range of function:
The function is invertible isn’t invertible because it doesn’t pass the horizontal line test doesn’t pass the vertical line test passes the vertical line test passes the horizontal line test .
Domain of inverse: Range of inverse:
Solution
The graph provided represents a periodic function that appears to be sinusoidal, given its wavelike structure.
Analysis:
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Domain of the function: The domain of this function includes all the x-values for which the function is defined. Since the graph extends infinitely in both directions along the x-axis, the domain is all real numbers.
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Range of the function: The range of the function is determined by the minimum and maximum y-values that the function reaches. From the graph, the function oscillates between a maximum value and a minimum value. If we estimate these values, the function appears to oscillate between approximately and , so the range is .
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Invertibility: To determine if the function is invertible, we use the horizontal line test. A function is invertible if every horizontal line intersects the graph at most once. However, this sinusoidal function intersects a horizontal line more than once, meaning the function is not invertible.
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Domain of the inverse: Since the function is not invertible, it does not have an inverse in the conventional sense. Therefore, the domain of the inverse does not exist.
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Range of the inverse: Similarly, because the function is not invertible, the range of the inverse does not exist.
Final Answer:
- Domain of the function: All real numbers
- Range of the function:
- The function is not invertible because it doesn't pass the horizontal line test.
- Domain of the inverse: Not applicable (since the function isn't invertible)
- Range of the inverse: Not applicable (since the function isn't invertible)
Would you like further details or have any questions? Here are some related questions you might find interesting:
- How can the range of a sinusoidal function be determined precisely?
- What does it mean for a function to be periodic, and how does it affect its domain and range?
- Can a sinusoidal function be made invertible? If so, how?
- What are some practical examples of periodic functions?
- How do the amplitude and period of a sinusoidal function affect its graph?
Tip: For functions that are not invertible in their entirety, you can restrict the domain to a portion where the function is one-to-one, making it possible to define an inverse.
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Math Problem Analysis
Mathematical Concepts
Functions
Invertibility
Horizontal Line Test
Domain
Range
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12
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