Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduced the concept of one-to-one functions and their inverses, emphasizing the importance of the horizontal line test to ensure a function is invertible. It detailed the process for finding an inverse—solving for x and interchanging variables—and demonstrated that the derivative of an inverse function is the reciprocal of the original function's derivative at corresponding points. Graphical interpretations, including reflections across the line y = x, solidify understanding of the relationship between a function and its inverse.

Learning Objectives

1

Define one-to-one functions and explain the concept of inverse functions.

2

Apply the horizontal line test to determine whether a function is one-to-one.

3

Learn to find inverses of functions by algebraically solving for x and then swapping variables.

4

Understand and use the derivative rule for inverse functions to compute slopes of inverse curves.

5

Interpret the graphical relationship between a function and its inverse, including reflection across the line y = x.

Key Concepts

CONCEPT

DEFINITION

One-to-One Function

A function where each element in the domain maps to a unique element in the range, meaning f(x1) ≠ f(x2) whenever x1 ≠ x2.

Inverse Function

A function that reverses the effect of the original function, such that f⁻¹(b) = a if f(a) = b; its domain and range are interchanged relative to the original function.

Horizontal Line Test

A graphical method used to determine if a function is one-to-one; a function is one-to-one if every horizontal line intersects its graph at most once.

Reflection across the line y = x

A method to obtain the graph of an inverse function by reflecting the graph of the original function over the line y = x.

Derivative Rule for Inverse Functions

If f is differentiable and one-to-one with f′(a) ≠ 0, then the derivative of its inverse at b = f(a) is given by (f⁻¹)′(b) = 1/(f′(a)).

Example Problems

Example 1

Which of the functions graphed are one-to-one, and which are not?

Example 2

Which of the functions graphed are one-to-one, and which are not?

Example 3

Which of the functions graphed are one-to-one, and which are not?

Example 4

Which of the functions graphed are one-to-one, and which are not?

Example 5

Which of the functions graphed are one-to-one, and which are not?
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Step-by-Step Explanations

QUESTION

Find the inverse of the function y = (1/2)x + 1.

STEP-BY-STEP ANSWER:

Step 1: Write the function as y = (1/2)x + 1.
Step 2: Solve for x in terms of y by subtracting 1 from both sides: y - 1 = (1/2)x.
Step 3: Multiply both sides by 2 to isolate x: x = 2(y - 1).
Step 4: Interchange x and y to express the inverse function in the proper form: y = 2(x - 1).
Final Answer: The inverse function is f⁻¹(x) = 2(x - 1).

Finding the Inverse of a Linear Function

QUESTION

Given the function f(x) = x² (with domain x ≥ 0), find the derivative of the inverse function f⁻¹ at a corresponding point.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the inverse of f(x) = x², x ≥ 0, is f⁻¹(x) = √x.
Step 2: Compute the derivative of f(x): f′(x) = 2x.
Step 3: For a point (a, b) where b = f(a), by the derivative rule for inverse functions, (f⁻¹)′(b) = 1/(f′(a)).
Step 4: For example, if a = 2 then b = 4 and f′(2) = 4, so (f⁻¹)′(4) = 1/4.
Final Answer: The derivative of the inverse function at x = 4 is 1/4.

Computing the Derivative of an Inverse Function

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Common Mistakes

  • Confusing the inverse function notation f?¹(x) with the reciprocal (1/f(x)).
  • Failing to restrict the domain of functions (such as x²) to ensure they are one-to-one before finding an inverse.
  • Not properly swapping x and y after solving for the variable, leading to an incorrect inverse function.
  • Neglecting to verify that f?(a) is nonzero, which is necessary for using the derivative rule for inverse functions.