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:

  • 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.

  • 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 y=2y = 2 and y=2y = -2, so the range is [2,2][-2, 2].

  • 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.

  • 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.

  • 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 (,)(-\infty, \infty)
  • Range of the function: [2,2][-2, 2]
  • 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:

  1. How can the range of a sinusoidal function be determined precisely?
  2. What does it mean for a function to be periodic, and how does it affect its domain and range?
  3. Can a sinusoidal function be made invertible? If so, how?
  4. What are some practical examples of periodic functions?
  5. 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