Question

What is the inverse of f (x)=10-×^2

Answer 1

Finding the inverse function of

$\mathsf{f(x)=10-x^2}$


Remember that when you compose $f$ with its inverse $\mathsf{f^{-1}},$ you'll get the identity function:

$\mathsf{(f\circ f^{-1})(x)=x}\\\\ \mathsf{f\big[f^{-1}(x)\big]=x}$


So if

$\mathsf{f(x)=10-x^2}$


then

$\mathsf{f\big[f^{-1}(x)\big]=10-[f^{-1}(x)]^2}\\\\ \mathsf{x=10-[f^{-1}(x)]^2}\\\\ \mathsf{[f^{-1}(x)]^2=10-x}\\\\ \mathsf{f^{-1}(x)=\pm \sqrt{10-x}}\\\\ \mathsf{f^{-1}(x)=-\sqrt{10-x}~~~or~~~f^{-1}(x)=\sqrt{10-x}}$


The sign of the inverse depends on the domain of $\mathsf{f(x)}$ itself, and where it's invertible.


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I hope this helps.


Tags: inverse function definition identity domain algebra