https://file.aiquickdraw.com/mathbot/file/1724762130659cg4cfr6n.jpg

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: 


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 \( y = 2 \) and \( y = -2 \), so the range is \( [-2, 2] \).

- **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**: \([-2, 2]\)
- **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.

Explore the concept of function invertibility and its implications on domain and range. This detailed analysis covers how to determine if a function is invertible using the horizontal line test. Learn about the domain and range of functions, illustrated with examples and visual aids.

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