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.....
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
- 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.