2018 international practice exam ab mcq, part b - calculator required\nselected values of the increasing…
2018 international practice exam ab mcq, part b - calculator required\nselected values of the increasing function h and its derivative h are shown in the table above.\nif g is a differentiable function such that h(g(x)) = x for all x, what is the value of g(7)?\na -1/10\nb 1/10\nc 1/5\nd 7/5
Answer
Explanation:
Step1: Recall inverse - function derivative formula
If (h(g(x))=x) for all (x), then (h) and (g) are inverse functions of each other. The formula for the derivative of an inverse function is (g^{\prime}(y)=\frac{1}{h^{\prime}(x)}), where (y = h(x)).
Step2: Find the (x) - value such that (h(x)=7)
From the table, when (x = 3), (h(3)=7).
Step3: Use the inverse - function derivative formula
We want to find (g^{\prime}(7)). Since (y = 7) and (h(3)=7), by the formula (g^{\prime}(y)=\frac{1}{h^{\prime}(x)}), we substitute (x = 3) into the formula. Given (h^{\prime}(3)=5), then (g^{\prime}(7)=\frac{1}{h^{\prime}(3)}).
Step4: Calculate the value of (g^{\prime}(7))
(g^{\prime}(7)=\frac{1}{5}).
Answer:
C. (1/5)