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

[elemis]

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 ?

 

[Tinyboss]

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.

 

[elemis]

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 ?

 

[disregardthat]

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.

 

[CellCoree]

I would like to clarify that the domain and range of a function and its inverse are not necessarily the same. In fact, they are typically different.

The domain of a function refers to the set of all possible input values for which the function is defined. The range, on the other hand, refers to the set of all possible output values that the function can produce. In other words, the domain and range are determined by the function itself.

When we take the inverse of a function, we are essentially swapping the input and output values. This means that the domain and range of the inverse function will be the range and domain of the original function, respectively. So, in general, the domain and range of the inverse function will be different from the original function.

However, in some cases, such as when the original function is a one-to-one function, the domain and range of the inverse function can be the same as the original function. This is because a one-to-one function has a unique input for every output, and vice versa. In such cases, the inverse function can be seen as a reflection of the original function over the line y=x, resulting in the same domain and range.

In summary, while it is possible for the domain and range of an inverse function to be the same as the original function, this is not always the case. It is important to carefully consider the nature of the original function and its inverse when determining their respective domains and ranges.