let f and g be inverse functions that are differentiable for all x. if f(-5)=7 and g(7)=3, which of the…

let f and g be inverse functions that are differentiable for all x. if f(-5)=7 and g(7)=3, which of the following statements must be false? i. f(3)= - 1/3 ii. f(-5)=1/3 iii. f(7)=1/3 a. i only b. ii only c. iii only d. i and iii only

let f and g be inverse functions that are differentiable for all x. if f(-5)=7 and g(7)=3, which of the following statements must be false? i. f(3)= - 1/3 ii. f(-5)=1/3 iii. f(7)=1/3 a. i only b. ii only c. iii only d. i and iii only

Answer

Explanation:

Step1: Recall the formula for the derivative of an inverse - function

If (y = f(x)) and (g(x)) is the inverse of (f(x)), then (g^{\prime}(y)=\frac{1}{f^{\prime}(x)}) where (y = f(x)).

Step2: Analyze statement I

Given (f(-5)=7) and (g^{\prime}(7) = 3). By the formula (g^{\prime}(y)=\frac{1}{f^{\prime}(x)}) with (y = 7) and (x=-5), we have (g^{\prime}(7)=\frac{1}{f^{\prime}(-5)}), so (f^{\prime}(-5)=\frac{1}{3}), and statement I (f^{\prime}(3)=-\frac{1}{3}) is false.

Step3: Analyze statement II

Since (f(-5)=7), then by the inverse - function derivative formula (g^{\prime}(7)=\frac{1}{f^{\prime}(-5)}). Given (g^{\prime}(7) = 3), we get (f^{\prime}(-5)=\frac{1}{3}), so statement II (f^{\prime}(-5)=\frac{1}{3}) is true.

Step4: Analyze statement III

We know that if (y = f(x)) and (g) is the inverse of (f), (g^{\prime}(y)=\frac{1}{f^{\prime}(x)}). There is no information to support the relationship in statement III (f^{\prime}(7)=\frac{1}{3}), and it is not consistent with the inverse - function derivative formula based on the given (f(-5)=7) and (g^{\prime}(7) = 3), so statement III is false.

Answer:

A. I only