To find the inverse function \( m^{-1}(x) \) from the given points on the function \( m(x) \), we need to switch the x and y coordinates of each point. The points given are:
- \( (127, -3) \)
- \( (1, 0) \)
- \( (3, 1) \)
- \( (81, 4) \)
Switching the coordinates, we get the points for the inverse function:
- \( (-3, 127) \)
- \( (0, 1) \)
- \( (1, 3) \)
- \( (4, 81) \)
Now, we can express these points as pairs of \( (y, x) \), where \( y = m^{-1}(x) \):
- \( m^{-1}(-3) = 127 \)
- \( m^{-1}(0) = 1 \)
- \( m^{-1}(1) = 3 \)
- \( m^{-1}(4) = 81 \)
Looking for patterns among the outputs:
From \( m^{-1}(0) = 1 \), we can say \( m^{-1}(x) \) might take the form of a constant function due to the nature of inversion. However, to find a general equation based on known logarithmic functions or polynomial forms, we will analyze the points given.
The points do not display a straightforward polynomial or exponential behavior. To fit them into a functional form, let's analyze the potential responses provided.
Upon evaluating the potential responses:
- \( m^{-1}(x) = 3^x \)
- \( m^{-1}(x) = 3x \)
- \( m^{-1}(x) = x^3 \)
- \( m^{-1}(x) = \log_3{x} \)
Given none of these forms fit seamlessly within the data set, particularly for matching each point derived earlier, the outlier points in combination can suggest a possible logarithmic relationship.
Looking at the closeness of outputs, I would advocate to explore logarithmic characteristics identified through the equations provided.
Ultimately, as no direct calculation can yield an appropriate linear fit, we can assert interest primarily around function transformations, concluding, the point structures and available options lead one to deduce:
\( m^{-1}(x) = \log_3{x} \) fitting the observations of logarithmic base at \(3\).
Thus, the correct response would be:
\( m^{-1}(x) = \log_3{x} \).