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This property is the core of the inverse function definition: the inverse function "undoes" the effect of the original function, and vice versa. If we compose a function with its inverse (in any order), we get the identity function .
Composition of with :
This holds for all in the domain of (which is the range of ).
Composition of with :
This holds for all in the domain of .
Example:
We know that if , its inverse is . Let's verify the composition property:
Both compositions result in , as expected.
If we find the inverse of an inverse function, we get back the original function.
This makes sense because the process of finding an inverse is a "reversal". If we reverse something twice, we return to the original state.
If we have a composition of two functions, both of which have inverses, the inverse of the composition is the composition of their inverses, but in reverse order.
Let and be two functions with inverses and . Then the inverse of the composition is:
Note the reversed order: is applied first, then .
Imagine putting on socks () and then shoes (). To undo this (the inverse), you must take off the shoes () first, then take off the socks (). The order is reversed.
Example:
Let (its inverse is ) and (its inverse is ).
Find :
Find the inverse of :
Let . Swap and : .
Find :
Since the results from steps and are the same, it is proven that .
The domain of the original function becomes the range of its inverse function , and the range of the original function becomes the domain of its inverse function .
Solve for : .
So, .
