- #1

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## Summary:

- Let say I have f(x) = ##-x^2 + 3 , x \leq 0##. I want to find the domain of composite function ##ff^{-1}## and ##f^{-1} f##

##f(x)=-x^2 + 3## so ##f^{-1} (x)=- \sqrt{3-x} ~, x \leq 3##

##ff^{-1} = - (- \sqrt{3-x})^2 + 3 = x## and the domain will be ##x \leq 3##

##f^{-1} f = - \sqrt{3-(-x^2+3)} = -x ## and the domain will be ##x \leq 0##

My question:

##ff^{-1} (x)## or ##f^{-1} f(x) ## is not always equal to ##x## and their domain can be different?

Thanks

##ff^{-1} = - (- \sqrt{3-x})^2 + 3 = x## and the domain will be ##x \leq 3##

##f^{-1} f = - \sqrt{3-(-x^2+3)} = -x ## and the domain will be ##x \leq 0##

My question:

##ff^{-1} (x)## or ##f^{-1} f(x) ## is not always equal to ##x## and their domain can be different?

Thanks