How does melodic inversion transform a fugue subject?

In the context of a fugue, melodic inversion is a technique used to elaborate on musical material by systematically reversing the direction of the intervals within a subject.12

When a subject is inverted, its original melodic contour is mirrored: ascending intervals are transformed into descending ones, and descending intervals are transformed into ascending ones. For example, if the original subject moves upward by a whole tone, the inverted version will move downward by a whole tone.12

This method allows composers to develop and manipulate the fugue's primary theme while maintaining a clear relationship to the original material. In some instances, such as in J.S. Bach's The Art of the Fugue, composers may create "mirror fugues," where a second fugue is composed as the exact mirror inversion of the first.12