QuestionFind a formula for the inverse of the function. $$\begin{array}{l} f(x)=\sqrt{x^{2}+2 x}, x>0 . \\ f^{-1}(x)= \end{array}$$

Studdy Solution
Verify the inverse function by checking if $f(f^{-1}(x)) = x$. Substitute $f^{-1}(x) = -1 + \sqrt{1 + x^2}$ into $f(x)$:
$$f(f^{-1}(x)) = \sqrt{((-1 + \sqrt{1 + x^2})^2 + 2(-1 + \sqrt{1 + x^2}))}$$
Simplify the expression:
$$= \sqrt{(1 - 2\sqrt{1 + x^2} + (1 + x^2) + 2(-1 + \sqrt{1 + x^2}))}$$ $$= \sqrt{(1 + x^2)}$$ $$= x$$
Thus, the inverse function is verified.
The formula for the inverse function is $f^{-1}(x) = -1 + \sqrt{1 + x^2}$.