/*! 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 132 Proof Prove that if \(f\) is a o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 132

Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

Short Answer

Expert verified
If \(f\) is a one-to-one odd function, its inverse \(f^{-1}\) is also an odd function.

Step by step solution

01

Declare that \(f\) is one-to-one and odd

Assume that \(f\) is a one-to-one odd function. This means that \(f(-x) = -f(x)\) for any \(x\) within the domain of \(f\), and if \(f(a) = f(b)\), then \(a = b\).
02

Analyze the odd function property of \(f^{-1}\)

The goal here is to show that \(f^{-1}\) holds the property of an odd function which is \(f^{-1}(-y) = -f^{-1}(y)\). It needs to be demonstrated that this statement is true for all \(y\) within the range of \(f\).
03

Use the definition of the inverse function

Substitute \(f(-x)\) for \(y\) in the formula \(f^{-1}(-y) = -f^{-1}(y)\). This gives \(f^{-1}(-f(-x)) = -f^{-1}(f(-x))\). Since \(f\) is one-to-one, where \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\), it can be seen here that \(f(-x)\) is replaced with y, hence you have \(f^{-1}(-y) = -x\).
04

Apply the property of the odd function

Use the definition of the odd function \(f(-x) = -f(x)\). Substituting \(f(x)\) for y in \(f^{-1}(-y) = -x\), we get \(f^{-1}(-f(x)) = -x\), which applying the property of the inverse function \(f^{-1}(f(x)) = x\) gives \(-x = -x\), satisfying the condition for the inverse to be an odd function.
05

State that \(f^{-1}\) is odd

From steps 3 and 4, it has been shown that \(f^{-1}\) holds the property that the negative of its input equals the output of the negative input. Therefore, \(f^{-1}\) is an odd function.

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!

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

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=x \sqrt{x-3}\) (a) \(f(3)\) (b) \(f(12)\) (c) \(f(6)\)

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=2 f(x)$$.

Find the difference quotient and simplify your answer. $$f(x)=x^{3}+x, \quad \frac{f(x+h)-f(x)}{h}, \quad h \neq 0$$

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=x \sqrt{4-x^{2}}$$

The depreciation \(D\) (in millions of dollars) of the WD-40 Company assets from 2009 through 2013 can be approximated by the function $$D(t)=1.9 \sqrt{t+3.7}$$,where \(t=0\) represents 2009.(a) Describe the transformation of the parent function \(f(t)=\sqrt{t}\). (b) Use a graphing utility to graph the model over the interval \(0 \leq t \leq 4\). (c) According to the model, in what year will the depreciation of WD-40 assets be approximately 6 million dollars? (d) Rewrite the function so that \(t=0\) represents 2011 . Explain how you got your answer.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.