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!