/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Classify the following statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 7

Classify the following statements as either true or false. The function \(f\) is the inverse of \(f^{-1}\).

Short Answer

Expert verified
True

Step by step solution

01

Understand Inverse Functions

To classify the statement, first recall the definition of inverse functions. If a function \(f\) has an inverse function \(f^{-1}\), then for any input \(x\), \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
02

Analyze the Given Statement

The statement says 'The function \(f\) is the inverse of \(f^{-1}\)'. This means we need to check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) hold true.
03

Verify the Statement

From the definitions of inverse functions, it is true that if \(f\) and \(f^{-1}\) are inverses of each other, then the statements \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) must hold. Therefore, the function \(f^{-1}\) is itself the inverse of \(f\).
04

Conclusion

Since \(f\) and \(f^{-1}\) satisfy the properties of inverse functions with respect to each other, the statement 'The function \(f\) is the inverse of \(f^{-1}\)' is true.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function
A function is a basic concept in mathematics. It describes a specific relationship between two sets: the domain (input values) and the range (output values). For each input value (often denoted as \(x\)), there is exactly one output value (often denoted as \(f(x)\)). Functions can be represented in various forms, such as equations, graphs, tables, or even words.

Here are some key points about functions:
  • They are written as \(f: X \rightarrow Y\), where \(X\) is the domain and \(Y\) is the range.
  • Every element in the domain maps to one and only one element in the range.
  • The notation \(f(x)\) indicates the value of the function at the input \(x\).
Understanding functions is crucial for grasping more complex concepts, such as inverse functions.
Inverse Function
Inverse functions reverse the operations of the original function. If you apply a function \(f\) and then apply its inverse \(f^{-1}\), you return to the original input value. Mathematically, this is represented as \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

To verify whether two functions are inverses, you must check these two conditions. When both conditions are satisfied, the functions undo each other鈥檚 effects.

Key aspects of inverse functions include:
  • If \(f\) is a function with domain \(X\) and range \(Y\), then \(f^{-1}\) will have domain \(Y\) and range \(X\).
  • Inverse functions essentially 'flip' the role of the domain and range.
  • Not all functions have inverses. Only one-to-one (bijective) functions, which have a unique output for every unique input, have inverses.
This concept is essential to solve problems like the one in the exercise, where confirming that \(f\) is the inverse of \(f^{-1}\) hinges on these properties.
Mathematical Properties
Mathematical properties help us analyze and solve expressions involving functions and their inverses. When working with inverse functions, some crucial properties to remember include:

  • Identity Property: For a function \(f\) and its inverse \(f^{-1}\), the equations \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) must hold true for all \(x\) in their respective domains.
  • Symmetry: The graphical representation of \(f\) and \(f^{-1}\) will be symmetric about the line \(y = x\).
  • Composition: Composing a function with its inverse (either way) yields the identity function, which leaves the input unchanged.
These properties are essential for validating the relationship between functions and their inverses. Understanding them will enable you to solve similar problems and deepen your comprehension of functions and their inverses.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The function $$ Y(x)=21.77 \ln \frac{x}{5.5} $$ can be used to estimate the number of years \(Y(x)\) after 1982 required for the world's humpback whale population to reach \(x\) thousand whales. a) In what year will the whale population reach \(15,000 ?\) b) In what year will the whale population reach \(25,000 ?\) c) Graph the function.

Simplify. \(t^{1 / 5} t^{2 / 3}\)

The number of computers infected by a virus \(t\) days after it first appears usually increases exponentially. In 2009 the "Conflicker" worm spread from about 2.4 million computers on January 12 to about 3.2 million computers on January \(13 .\) Data: PC World a) Find the exponential growth rate \(k\) and write an equation for an exponential function that can be used to predict the number of computers infected \(t\) days after January \(12,2009\) b) Assuming exponential growth, estimate how long it took the Conflicker worm to infect 10 million computers.

As of May 2016 , the highest price paid for a painting was 300 million dollars, paid in 2015 for Willem de Kooning's "Interchange." The same painting was purchased for 20.6 million dollars in \(1989 .\) Data: wsj. com, \(2 / 25 / 16\) a) Find the exponential growth rate \(k,\) and determine the exponential growth function that can be used to estimate the painting's value \(V(t),\) in millions of dollars, \(t\) years after \(1989 .\) b) Estimate the value of the painting in 2025 c) What is the doubling time for the value of the painting? d) How many years after 1989 will it take for the value of the painting to reach 1 billion dollars?

Find each of the following, given that $$ f(x)=\frac{1}{x+2} \quad \text { and } \quad g(x)=5 x-8 $$ The domain of \(f / g\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.