Book cover for Algebra and Trigonometry

Algebra and Trigonometry

Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger

ISBN #9780321693983

4th Edition

5,839 Questions

Group icon
199,024 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on the concept of inverse relations and inverse functions, emphasizing how to obtain an inverse by interchanging the x and y coordinates and then solving for the new dependent variable. It reinforces the importance of one-to-one functions, highlighted through graphical tests like the horizontal-line test. The section also demonstrates how inverses are confirmed using composition of functions and addresses techniques such as domain restriction in cases where the inverse is not naturally a function.

Learning Objectives

1

Explain the concept of inverse relations and what distinguishes an inverse function from a general relation.

2

Demonstrate the process of finding an inverse function by interchanging the variables and solving for the new dependent variable.

3

Identify and apply the horizontal-line test and vertical-line test to determine if a function is one-to-one and if its inverse is also a function.

4

Use composition of functions to verify inverses and understand the significance of one-to-one functions in establishing inverses.

5

Apply the procedure for restricting domains to ensure that non one-to-one functions can have inverses that are functions.

Key Concepts

CONCEPT

DEFINITION

Inverse Relation

A relation formed by switching the roles of inputs and outputs in the original relation. When the inverse is a function, every output from the original function becomes the unique input for the inverse.

Inverse Function

A function denoted by f⁻Âč such that f⁻Âč(f(x)) = x for every x in the domain of f, meaning that it 'undoes' the action of f. Its graph is the reflection of the graph of f across the line y = x.

One-to-One Function

A function where different inputs produce different outputs. Mathematically, if f(a) = f(b) then a must equal b. Only one-to-one functions have inverses that are functions.

Horizontal-Line Test

A graphical test used to determine if a function is one-to-one. If any horizontal line intersects the graph more than once, then the function is not one-to-one.

Restricting the Domain

Limiting a function’s domain to ensure that it becomes one-to-one so that its inverse can also be defined as a function.

Example Problems

Example 1

Find the inverse of the relation. $$\{(7,8),(-2,8),(3,-4),(8,-8)\}$$

Example 2

Find the inverse of the relation. $$\{(0,1),(5,6),(-2,-4)\}$$

Example 3

Find the inverse of the relation. $$\{(-1,-1),(-3,4)\}$$

Example 4

Find the inverse of the relation. $$\{(-1,3),(2,5),(-3,5),(2,0)\}$$

Example 5

Find an equation of the inverse relation. 2 x^{2}+5 y^{2}=4
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

How do you find the inverse of the function f(x) = 2x - 3?

STEP-BY-STEP ANSWER:

Step 1: Write the function as y = 2x - 3.
Step 2: Interchange x and y to obtain x = 2y - 3.
Step 3: Solve for y by adding 3 to both sides: x + 3 = 2y.
Step 4: Divide both sides by 2: y = (x + 3)/2.
Final Answer: The inverse function is f⁻Âč(x) = (x + 3)/2.

Finding the Inverse of a Function

QUESTION

How do you determine if a function is one-to-one using the horizontal-line test?

STEP-BY-STEP ANSWER:

Step 1: Graph the function on the coordinate plane.
Step 2: Draw horizontal lines across the entire range of the graph.
Step 3: Check if any horizontal line intersects the graph at more than one point.
Step 4: If every horizontal line intersects the graph at most once, then the function is one-to-one.
Final Answer: If the horizontal-line test is passed, the function is one-to-one and its inverse will also be a function.

Testing a Function for One-to-One

Scroll left
Scroll right

Common Mistakes

  • Confusing the notation f?Âč(x) with a reciprocal of f(x); f?Âč(x) is not 1/f(x).
  • Failing to properly interchange the variables before solving for the new y.
  • Overlooking the necessity of the function being one-to-one, which can lead to an inverse that is not a function.
  • Improperly applying the horizontal-line test, such as misinterpreting intersections caused by points of tangency.
  • Neglecting to restrict the domain when a function is not one-to-one naturally.