Question - Finding the Inverse Function of a Given Function
Solution:
To find $$ f^{-1}(x) $$, the inverse of the function $$ f(x) = \sqrt{x} - 10 $$, follow these steps:1. Replace $$ f(x) $$ with $$ y $$:\[ y = \sqrt{x} - 10 \]2. Swap $$ x $$ and $$ y $$ to find the inverse:\[ x = \sqrt{y} - 10 \]3. Solve for $$ y $$:\[ x + 10 = \sqrt{y} \]Now square both sides to get rid of the square root:\[ (x + 10)^2 = (\sqrt{y})^2 \]\[ x^2 + 20x + 100 = y \]The inverse function $$ f^{-1}(x) $$ is then:\[ f^{-1}(x) = x^2 + 20x + 100 \]
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