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

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Discussion Overview

The discussion revolves around the properties of domains and ranges for inverse functions, particularly when the inverse function is proposed to have the same equation as the original function. Participants explore whether the domain and range of a function and its inverse can be the same under certain conditions.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions if the domain and range of a function f(x) will be the same as its inverse f-1(x) if they share the same equation.
  • Another participant states that generally, the domain of f is the range of f-1 and vice versa, arguing that if f has the same equation as its inverse, the ranges do not have to be the same as the domains, using f(x) = -x as a counterexample.
  • A later reply clarifies the initial question by providing specific conditions for f(x), suggesting that if the range is f(x) > 3 and the domain is x > 0, it questions whether f-1(x) would also have the same domain and range.
  • Another participant asserts that the previous response understood the question correctly and reiterates the counterexample provided.

Areas of Agreement / Disagreement

Participants express differing views on whether the domain and range can be the same for a function and its inverse when the function's equation is identical to that of its inverse. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants have not fully explored the implications of restricting the domain of f(x) or the specific conditions under which the inverse function's properties might change. There are also assumptions about the nature of the functions that have not been explicitly stated.

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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