Question

let ( g(x)=3 - 2x - x^{3} ) and let ( f ) be the inverse function of ( g ). assume that ( g(1)=0 ). ( f(0)= )

Explanation:

Step1: Use the property of inverse functions

If \(f\) is the inverse function of \(g\), then \(f(g(x)) = x\) for all \(x\) in the domain of \(g\) and \(g(f(y))=y\) for all \(y\) in the domain of \(f\). Also, we know that if \(y = f(x)\), then \(x = g(y)\) and \(f^{\prime}(x)=\frac{1}{g^{\prime}(y)}\). Given \(g(1) = 0\), then \(f(0)=1\).

Step2: Find the derivative of \(g(x)\)

Differentiate \(g(x)=3 - 2x - x^{3}\) with respect to \(x\). Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(g^{\prime}(x)=-2-3x^{2}\).

Step3: Evaluate \(g^{\prime}(1)\)

Substitute \(x = 1\) into \(g^{\prime}(x)\). So \(g^{\prime}(1)=-2-3(1)^{2}=-2 - 3=-5\).

Step4: Use the formula for the derivative of an inverse function

Since \(f^{\prime}(a)=\frac{1}{g^{\prime}(f(a))}\), when \(a = 0\), and \(f(0)=1\), then \(f^{\prime}(0)=\frac{1}{g^{\prime}(f(0))}=\frac{1}{g^{\prime}(1)}\).

Answer:

\(-\frac{1}{5}\)