Math Problem Statement

The table above gives selected values for a differentiable and increasing function f and its derivative f'. If g(x) = f⁻¹(x) for all x, which of the following is a correct expression for g'(0)?

Solution

We are given a table for a differentiable and increasing function f(x)f(x) and its derivative f(x)f'(x). We are asked to find g(0)g'(0), where g(x)=f1(x)g(x) = f^{-1}(x) is the inverse function of f(x)f(x).

Key Information:

  1. g(x)=f1(x)g(x) = f^{-1}(x)
  2. The derivative of the inverse function can be expressed as: g(x)=1f(g(x))g'(x) = \frac{1}{f'(g(x))}
  3. We are asked to find g(0)g'(0), meaning we need to find g(0)g'(0).

Steps:

  1. Since g(x)=f1(x)g(x) = f^{-1}(x), we know f(g(x))=xf(g(x)) = x.
  2. To find g(0)g'(0), first locate the value of ff such that f(g(0))=0f(g(0)) = 0. From the table, f(0)=3f(0) = 3, so g(3)=0g(3) = 0, meaning g(0)=4g(0) = -4.
  3. Now, use the formula for the derivative of the inverse: g(0)=1f(g(0))=1f(4)g'(0) = \frac{1}{f'(g(0))} = \frac{1}{f'(-4)}
  4. From the table, f(4)=1f'(-4) = 1, so: g(0)=11=1g'(0) = \frac{1}{1} = 1

Thus, the correct answer is option C.


Would you like any further clarifications or details?

Here are 5 related questions to expand your understanding:

  1. How do you compute the derivative of an inverse function in general?
  2. What are the conditions for a function to have an inverse?
  3. How does the relationship between a function and its inverse affect their derivatives?
  4. Can this method of finding the derivative of an inverse be applied to non-differentiable functions?
  5. What would change in the calculation if the function f(x)f(x) were not increasing?

Tip: When working with inverse functions, always check how the function and its inverse are related through specific values, especially at critical points.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Derivatives
Chain Rule

Formulas

g'(x) = 1 / f'(g(x))

Theorems

Derivative of an inverse function

Suitable Grade Level

Grades 11-12