Question

the table above gives selected values for a differentiable and increasing function (f) and its derivative. if (g(x)=f^{-1}(x)) for all (x), which of the following is a correct expression for (g^{prime}(0))?
a (g^{prime}(0)=f^{prime}(0)=2)
b (g^{prime}(0)=\frac{1}{f^{prime}(0)}=\frac{1}{2})
c (g^{prime}(0)=\frac{1}{f^{prime}(-4)} = 1)
d (g^{prime}(0)=-\frac{f^{prime}(0)}{(f(0))^{2}}=-\frac{2}{9})
answer b
incorrect. this response comes from thinking that because (f) and (g) are inverses

Explanation:

Step1: Recall inverse - function derivative formula

If \(g(x)=f^{-1}(x)\), then \(g^{\prime}(x)=\frac{1}{f^{\prime}(g(x))}\). We want to find \(g^{\prime}(0)\), so we need to find the value of \(x\) such that \(f(x) = 0\).

Step2: Find the \(x\) - value for which \(f(x)=0\)

From the table, when \(x=-4\), \(f(-4)=0\). Since \(g(x)=f^{-1}(x)\), then \(g(0)=-4\) (because if \(y = f(x)\) and \(x = g(y)\), when \(y = 0\), \(x=-4\) according to the table).

Step3: Calculate \(g^{\prime}(0)\)

Substitute \(x = 0\) into the formula \(g^{\prime}(x)=\frac{1}{f^{\prime}(g(x))}\). Since \(g(0)=-4\), then \(g^{\prime}(0)=\frac{1}{f^{\prime}(g(0))}=\frac{1}{f^{\prime}(-4)}\). From the table, \(f^{\prime}(-4)=1\), so \(g^{\prime}(0)=\frac{1}{1}=1\).

Answer:

C. \(g^{\prime}(0)=\frac{1}{f^{\prime}(-4)} = 1\)