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The problem states that Kaleb incorrectly determined the inverse of the function \( y = 4 - \sqrt{-x} \) to be \( y = -(x - 4)^2 \), and used a graphing calculator to obtain a parabola. The task is to explain why the graph of the correct inverse is not a complete parabola.

### Explanation:

1. **Original Function**: The function \( y = 4 - \sqrt{-x} \) involves a square root, which defines a restricted range for \( y \) and a restricted domain for \( x \). Specifically:
   - \( \sqrt{-x} \) means that \( x \) must be less than or equal to 0, i.e., \( x \leq 0 \).
   - The square root function only outputs non-negative values, so the output of \( 4 - \sqrt{-x} \) will be less than or equal to 4 but greater than or equal to 0. 

2. **Incorrect Inverse**: Kaleb’s inverse \( y = -(x - 4)^2 \) describes a parabola that opens downwards. However, a parabola represents a complete curve, meaning it extends both in the positive and negative \( x \)-directions without restriction.

3. **Why the Correct Inverse is Not a Complete Parabola**:
   - The correct inverse of \( y = 4 - \sqrt{-x} \) will involve isolating \( x \) in terms of \( y \), which gives a restricted range. 
   - Taking the inverse of a square root function typically results in a relation where only part of the parabola is defined. Specifically, only one branch of the parabola would be valid (either the positive or negative side), depending on how the square root is defined in the original function. In this case, the inverse of the square root function would likely only produce the "top half" or "bottom half" of the parabola, not a full parabola.

4. **Conclusion**: The correct inverse is not a complete parabola because the domain and range of the original function restrict the inverse function to only a part of the full parabola. Kaleb’s error was assuming the inverse should be a full parabola, but in reality, only a portion of the parabola is valid for the inverse function.

Would you like a more detailed walkthrough of how to find the correct inverse of the original function?

### Relative Questions:
1. How do you find the inverse of a square root function in general?
2. What does the graph of \( y = 4 - \sqrt{-x} \) look like, and how does that relate to its inverse?
3. Why does the square root function have a restricted domain?
4. What is the graphical significance of taking the inverse of a function?
5. How can you use a graphing calculator to visualize partial curves?

**Tip**: Always check the domain and range of both the original function and its inverse to ensure that your inverse function makes sense graphically.

Kaleb incorrectly determined the inverse of y = 4 - √-x to be y = -(x - 4)^2 and used the graphing calculator to obtain a parabola. Explain why the graph of the correct inverse is not a complete parabola.

This problem explores why the inverse of the square root function y = 4 - √-x is not a complete parabola. Learn about the properties of inverse functions, square root functions, and how to correctly find the inverse in this step-by-step explanation. Suitable for high school students in Grades 10-12.

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