- Understand that \(f^{-1}(-2) = x\) implies we need to find the \(x\) for which \(f(x) = -2\).
- Locate the output \(-2\) in the provided table of values for \(f(x)\).
- Identify that when \(f(x) = -2\), the input \(x\) is \(1\).
- Conclude that the value of \(f^{-1}(-2)\) is \(\boxed{1}\).
Explanation
- Understanding Inverse Functions
To find the value of \(f^{-1}(-2)\), we must first understand the relationship between a function and its inverse. By definition, if \(f(x) = y\), then the inverse function satisfies \(f^{-1}(y) = x\). In this problem, we are given a table of values for the function \(f(x)\) and we are asked to find \(f^{-1}(-2)\). This means we are looking for the value of \(x\) such that \(f(x) = -2\).
- Finding the Value in the Table
We examine the provided table of values to find the row where \(f(x) = -2\):
| \(x\) | \(f(x)\) |
| :--- | :--- |
| \(-2\) | \(8\) |
| \(-1\) | \(5.5\) |
| \(-0.5\) | \(4\) |
| \(0\) | \(0.5\) |
| \(0.5\) | \(-1\) |
| \(1\) | \(-2\) |
| \(2\) | \(-7\) |
Looking at the table, we locate the value \(-2\) in the \(f(x)\) column. In the corresponding row, we see that the value of \(x\) is \(1\).
- Conclusion
Since the table shows that \(f(1) = -2\), it follows directly from the definition of the inverse function that \(f^{-1}(-2) = 1\). This matches the fourth option provided in the problem.
Examples
Inverse functions are widely used in daily life to "undo" operations. For example, if you know the price of an item including tax, you can use an inverse function to calculate the original pre-tax price. If \(f(x) = 1.05x\) represents the final price with a 5% tax, then \(f^{-1}(y) = y / 1.05\) allows you to work backwards from the final price \(y\) to the original price \(x\). This is exactly how cash registers reverse tax calculations to track net sales.