Understanding Domains and Ranges for Inverse Functions - A Noob's Guide

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The discussion centers on the relationship between the domains and ranges of a function and its inverse, particularly when they share the same equation. It is established that if f(x) and its inverse f-1(x) have the same equation, their domains and ranges do not necessarily have to match. An example provided is f(x) = -x, where restricting the domain affects the range of the inverse. The key takeaway is that the domain of f becomes the range of f-1 and vice versa, but this does not imply equality in specific cases. Ultimately, the domains and ranges can differ even when the functions are equivalent.
elemis
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Supposing I have a function f(x).

Let us suppose that f-1(x) has the same equation as f(x).

Will the domain and range as defined for f(x) be the same as for the inverse ?
 
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In general, if f-1 is the inverse of f, then the domain of f is the range of f-1, and vice-versa. So your question becomes, "if f has the same equation as its inverse, then does its range have to be the same as its domain?" The answer to that is "no". Consider f(x)=-x, and try restricting the domain of f to some subset of the real numbers.
 
Tinyboss said:
In general, if f-1 is the inverse of f, then the domain of f is the range of f-1, and vice-versa. So your question becomes, "if f has the same equation as its inverse, then does its range have to be the same as its domain?" The answer to that is "no". Consider f(x)=-x, and try restricting the domain of f to some subset of the real numbers.

No, you misunderstood the question I posed.

Lets us say that the range for f(x) is f(x) > 3 and its domain is x>0

Now let's say f-1(x) has the same equation as f(x).

Will f-1(x) have the same range and domain as f(x) ? That is, will the domain be x>0 and the range f-1(x)>3 ?
 
elemis said:
Will f-1(x) have the same range and domain as f(x) ? That is, will the domain be x>0 and the range f-1(x)>3 ?

Tinyboss understood you perfectly, and gave a counterexample.
 

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