https://file.aiquickdraw.com/mathbot/file/1724761976190nugt6i4z.jpg

Domain of function:
Range of function:

The function
is invertible
because it
passes the horizontal line test
.
Domain of inverse:
Range of inverse:

The graph provided appears to be a linear function. To determine the domain, range, and the invertibility of the function, let's analyze the graph.

### Analysis:
- **Domain of the function**: The domain of a function refers to all the possible input values (x-values). Since this is a linear function with no restrictions (the graph extends infinitely in both directions along the x-axis), the domain is all real numbers.
  
- **Range of the function**: The range of a function refers to all the possible output values (y-values). Similar to the domain, because the graph is linear and extends infinitely along the y-axis, the range is also all real numbers.
  
- **Invertibility**: A function is invertible if it passes the horizontal line test, meaning that any horizontal line drawn through the graph intersects it at most once. Since the graph represents a linear function, it does pass the horizontal line test, so the function is invertible.

- **Domain of the inverse**: The domain of the inverse function is the same as the range of the original function. Since the range of the original function is all real numbers, the domain of the inverse function is also all real numbers.
  
- **Range of the inverse**: The range of the inverse function is the same as the domain of the original function. Since the domain of the original function is all real numbers, the range of the inverse function is also all real numbers.

### Final Answer:
- **Domain of the function**: All real numbers \((-\infty, \infty)\)
- **Range of the function**: All real numbers \((-\infty, \infty)\)
- **The function is invertible** because it passes the horizontal line test.
- **Domain of the inverse**: All real numbers \((-\infty, \infty)\)
- **Range of the inverse**: All real numbers \((-\infty, \infty)\)

Would you like to know more details or have any questions? Here are some related questions you might find interesting:

1. How can we formally determine if a function is invertible?
2. What are the properties of a linear function?
3. How do you calculate the inverse of a linear function?
4. What is the horizontal line test and how does it relate to invertibility?
5. Can a function have an inverse if it’s not one-to-one?

**Tip:** When analyzing functions graphically, always consider their behavior at the extremes (as \(x \to \pm \infty\)) to determine domain and range effectively.

Explore the domain, range, and invertibility of functions using the horizontal line test. This detailed explanation covers how to determine if a function is invertible, including examples of functions passing the horizontal line test. Suitable for high school students studying advanced functions.

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