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The image shows a mathematical question about finding the inverse of a function based on a graph of a quadratic function and four multiple-choice options. The graph is of the parabola \( f(x) = x^2 - 5x - 4 \). The problem asks how to determine the inverse function and what the correct choice for its inverse is.

### Key Steps to Solve:

1. **Understand the function**: The given quadratic function \( f(x) = x^2 - 5x - 4 \) is not one-to-one over its entire domain, meaning it does not have a unique inverse. For an inverse to exist, the function must be restricted to one of its branches (either increasing or decreasing).

2. **Check the vertex**: The vertex of the parabola occurs at \( x = \frac{-(-5)}{2(1)} = \frac{5}{2} \), or \( x = 2.5 \). The function opens upwards, meaning it decreases to the left of \( x = 2.5 \) and increases to the right.

3. **Finding the inverse**: To find the inverse, we would solve the equation \( y = x^2 - 5x - 4 \) for \( x \) in terms of \( y \). However, the quadratic function needs to be restricted for the inverse to exist. Typically, for the right-hand side (increasing branch), the inverse will involve the positive square root, and for the left-hand side (decreasing branch), the inverse will involve the negative square root.

4. **Answer options**: The options provided seem to present different solutions for the inverse using square roots, restricted to specific domains.

Would you like a detailed explanation of how to solve for the inverse algebraically, or should I guide you through choosing the correct multiple-choice option based on the graph? Let me know!

### Related Questions:
1. What is the method for finding the inverse of a quadratic function?
2. How does the vertex of a parabola help in determining the inverse?
3. Why does a quadratic function not have an inverse over its entire domain?
4. What are the steps to solve for \( x \) in terms of \( y \) in a quadratic equation?
5. How do you determine whether to use the positive or negative square root for the inverse function?

### Tip:
When finding the inverse of a function graphically, always check if the function is one-to-one and restrict its domain if needed!

Consider the function f(x) = x^2 - 5x - 4 shown in the graph. How can you restrict the domain so that f(x) is one-to-one? What is the equation of the inverse function?

Explore how to find the inverse of the quadratic function f(x) = x^2 - 5x - 4 by restricting its domain to ensure it's one-to-one. Learn key algebraic concepts and methods suitable for high school students.

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