Best Answer
Introduction
The inverse function of a function, denoted as f^-1(x), is a mathematical operation that reverses the original function. In other words, if f(x) = y, then f^-1(y) = x. The inverse function is only defined if the original function is one-to-one, meaning that each input value corresponds to a unique output value.
The Graph of z(x)
The graph of z(x) is not provided in the context, so we cannot directly determine the value of z^-1(-2) from the graph. However, we can discuss the general properties of inverse functions and how they relate to the graph of the original function.
Properties of Inverse Functions
The inverse function of a one-to-one function is also one-to-one.
The graph of an inverse function is a reflection of the graph of the original function over the line y = x.
The domain of the inverse function is the range of the original function, and vice versa.
The range of the inverse function is the domain of the original function, and vice versa.
Determining z^-1(-2) from the Graph
If we had the graph of z(x), we could use the properties of inverse functions to determine the value of z^-1(-2).
First, we would need to check if z(x) is one-to-one. This can be done by examining the graph and verifying that there are no horizontal lines that intersect the graph more than once.
If z(x) is one-to-one, then its inverse function exists.
To find z^-1(-2), we would locate the point (-2, z(-2)) on the graph of z(x).
The inverse function would then be the reflection of z(x) over the line y = x, so the point (-2, z(-2)) would be reflected to the point (z(-2), -2).
Thus, z^-1(-2) would be equal to z(-2).
Conclusion
Without the graph of z(x), we cannot directly determine the value of z^-1(-2). However, we have discussed the general properties of inverse functions and how they relate to the graph of the original function. If we were provided with the graph, we could use these properties to find the value of z^-1(-2) by locating the corresponding point on the graph of the inverse function.
Best Answer
z^-1(-2) = 5
To find the value of z^-1(-2), we need to determine the input value in the inverse function that results in an output of -2. Looking at the graph of z(x), we can see that when x = 5, z(x) = -2. Therefore, the value of z^-1(-2) is 5. So, when the output of z(x) is -2, the corresponding input value in the inverse function z^-1(x) is 5.
