Suppose that for solving for the inverse of a function that f(x) can be manipulated into the form of . Then, h(x) is inverted to show

. Afterwards, the function is reverted to show . Does this correctly show that h(g(x)) is in fact the inverse of f(x)?

No, it is not valid (or only trivially).

First some terminology.....

The

**domain** of a function is the set of all those elements that the function acts upon. Each element in the set is called an

**argument** for the function

The

**codomain** - or

**range** - of a function is the set of all elements that are the "output" of the function. So for any particular argument the element in the codomain is called the

**image** of the argument under the function.

So, Rule 1 for function...No element in the domain may have multiple images in the codomain.

Rule 2 for functions.....Functions are composed Right-to-Left

Rule 3 for functions.......Functions can be composed if and only if the codomain of a function is the domain of the function that follows it (i.e. as written, is "on the Left)

Rule 4 for functions....... For some image in the codomain, the

**pre-image set** is all those elements in the domain that "generate" this image. Notice that, although images are always single element, pre-images are

*sets* - although they may be sets with a single member in which case the function is said to have an inverse - not otherwise.

Look closely at what you wrote above. and check how many of these rules are violated.