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I often see that books define a function f as $f: X\rightarrow Y$. Now, I thought that the inverse is defined as $f^{-1}: Y\rightarrow X$. Then, in many places where I see the inverse mentioned, they write it as $f^{-1}(x)=y$ (assuming here that $Y$ is the image. That makes their domain equivalent. I thought that the inverse function's domain is the original function's range/image? Heck, the way that one finds an inverse in early algebra is to swap x and y, rearrange to get a function of y, then re-flip the variables.

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When using the generic letters x and y, convention is to let x be the independent variable of whatever function (original or inverse) under discussion. So when you switch the x and y, the set of replacement values for x and y (domain and range) also switch. If you don't want to "switch" the variables (as in application problems), I recommend that you use letters more meaningful to the situation (for example, T for temperature and P for pressure) and avoid the x and y notation which has independent and dependent connotations.

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