Question

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

Explanation:

Step1: Recall inverse - function derivative formula

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 \(g(x)\) is the inverse of \(f(x)\), then \(g(7)= - 5\). By the formula \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\), when \(y = 7\) and \(x=-5\), \(g^{\prime}(7)=\frac{1}{f^{\prime}(-5)}\). If \(f^{\prime}(-5)=\frac{1}{2}\), then \(g^{\prime}(7) = 2
eq-\frac{1}{2}\), so statement I is false.

Step3: Analyze statement II

We know that \(f^{\prime}(3)=-\frac{1}{2}\). Since \(g(x)\) is the inverse of \(f(x)\), if \(y = f(3)\) and \(x = 3\), then \(g^{\prime}(f(3))=\frac{1}{f^{\prime}(3)}=-2
eq\frac{1}{2}\), so statement II is false.

Step4: Analyze statement III

Given \(g(7)=3\) (from \(f(-5)=7\) and inverse - relationship), and by the formula \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\) where \(y = g(7)\) and \(x\) such that \(f(x)=g(7)\). If \(g(7)=3\), then \(f^{\prime}(x)\) corresponding to \(y = 3\) should satisfy \(g^{\prime}(7)=\frac{1}{f^{\prime}(x)}\). If \(g^{\prime}(7)=\frac{1}{3}\), then \(f^{\prime}(x)=3
eq\frac{1}{3}\), so statement III is false.

Answer:

All of the statements (I, II, and III) are false.