Consider the function defined on by

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Consider the function gg defined on [0,∞)[0,\infty) by g(x)=x+xg(x)=\sqrt{x}+x. The function is differentiable and strictly increasing on its domain, and thus possesses an inverse g−1g^{-1}. (a) Show that the equation g(x)=3g(x)=3 has exactly one solution on [0,∞)[0,\infty) and find that solution. (b) Determine g−1 ′(3)\displaystyle g^{-1}\,'(3). Express your answer in exact form, not a decimal approximation.

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