the table above gives selected values for a differentiable and increasing function (f) and its derivative…

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))?\na (g^{prime}(0)=f^{prime}(0)=2)\nb (g^{prime}(0)=\frac{1}{f^{prime}(0)}=\frac{1}{2})\nc (g^{prime}(0)=\frac{1}{f^{prime}(-4)} = 1)\nd (g^{prime}(0)=-\frac{f^{prime}(0)}{(f(0))^{2}}=-\frac{2}{9})\nanswer b\nincorrect. this response comes from thinking that because (f) and (g) are inverses

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))?\na (g^{prime}(0)=f^{prime}(0)=2)\nb (g^{prime}(0)=\frac{1}{f^{prime}(0)}=\frac{1}{2})\nc (g^{prime}(0)=\frac{1}{f^{prime}(-4)} = 1)\nd (g^{prime}(0)=-\frac{f^{prime}(0)}{(f(0))^{2}}=-\frac{2}{9})\nanswer b\nincorrect. this response comes from thinking that because (f) and (g) are inverses

Answer

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)