Find inverse functions for simple invertible functions, including rational examples.
Verify that two functions are inverse functions using composition
Problem
Verify by \(\text{both}~\text{compositions}\) that \(f(x)~=~3x~-~2\) and \(g(x)~=~(x~+~2)/3\) are inverse functions.
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What this problem is really about
Inverse functions must undo one another in both directions, so verify both compositions rather than stopping after one successful simplification. Substitute the entire inner function as a grouped input, simplify carefully, and check that each order returns the starting variable on its relevant domain. Matching the two compositions to each other is not enough unless both are the identity.
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