If $f(x) = x$ , Does the inverse of $f$, $f^{–1}$ exist? According to the definition, $f^{–1}= y$. But as it doesn't make sense I am going to assume it doesn't?
Does this function f(x) have an inverse?
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algebra-precalculus
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0The function inverse is itself. $f(x) =f^{-1}(x)$ – 2017-02-04
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0What definition are you using, how does it come up with $f^{-1}(x)=y,$ and why do you think this doesn't make sense? – 2017-02-04
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0Would it be okay to state f -1(x) = x or f -1(x) = y? – 2017-02-04
3 Answers
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This is the identity function. The inverse of the identity is itself. Hence, if $f(x)=x$, then $f^{-1}(y)=y$.
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The inverse is itself f(x) is y=x so even if you switch y and x, it's still y=x Also the graphs of inverses are symmetrical over the y=x line so if your function is that line then the inverse is itself
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In a nutshell, the functions are symmetrical. Taking the inverse of $f(x)$ will still yield x. To learn more in depth about why this holds, check the link below. https://en.wikipedia.org/wiki/Inverse_function