How would I solve the domain of the inverse for f(x) (given f(x)'s domain).

Dec 2018
29
2
USA
I have the equation f(x) = Capture.JPG, and its domain is Capture.JPG.

I solved for its inverse, Capture.JPG.

The domain for f-1(x) is originally x is less than or equal to 3, but with f(x)'s given domain, how do I solve for the new domain of f-1(x)?

Can someone guide me through the steps needed to solve such a problem? (Nerd)
 

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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
The domain of \(\displaystyle f^{-1}\) is the range of f. (Going from f to \(\displaystyle f^{-1}\) swaps "domain" and "range".) f(0)= 6 and as x goes to infinity, f(x) goes to negative infinity. So the domain of \(\displaystyle f^{-1}\) is all number less than or equal to 6.

(The inverse function to \(\displaystyle -x^2+ 6]\) is \(\displaystyle \sqrt{-x+ 6}\), not \(\displaystyle \sqrt{-x+ 3}\). Was that a typo?)
 
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