/*! 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 54 Each of the following functions ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 54

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=\frac{x^{9}}{4}\)

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \sqrt[9]{4x}\).

Step by step solution

01

Understand the Function

We are given the function \(f(x) = \frac{x^9}{4}\). This function is one-to-one, which means it has an inverse. Our goal is to find this inverse, denoted \(f^{-1}(x)\).
02

Interchange x and y

First, replace \(f(x)\) with \(y\) such that \(y = \frac{x^9}{4}\). To find the inverse, we swap \(x\) and \(y\), resulting in the equation \(x = \frac{y^9}{4}\).
03

Solve for y

To solve for \(y\), multiply both sides of the equation by 4 to isolate the \(y^9\) term: \(4x = y^9\).
04

Simplify to Find the Inverse

To isolate \(y\), take the ninth root of both sides: \(y = \sqrt[9]{4x}\). Thus, the inverse function can be expressed as \(f^{-1}(x) = \sqrt[9]{4x}\).
05

Express the Solution

The inverse function of \(f(x) = \frac{x^9}{4}\) is expressed as \(f^{-1}(x) = \sqrt[9]{4x}\) in proper notation.

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.

One-to-One Functions
A function is considered "one-to-one" if each input is paired with exactly one unique output, and each output is paired with exactly one unique input. This is a crucial characteristic for a function to have an inverse. If two different outputs come from the same input, it's not one-to-one and an inverse cannot be defined uniquely. Recognizing one-to-one functions can be achieved using several methods:
  • **Horizontal Line Test**: Draw horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function is not one-to-one.
  • **Checking Algebraically**: Ensure for a function \(f(x)\), if \(f(a) = f(b)\) then \(a = b\) must hold true for all input values \(a\) and \(b\).
Thorough understanding of whether a function is one-to-one or not is important. It's the first step to determine if the function can have an inverse.
Finding Inverses
To find the inverse of a function is like turning it into its opposite. This involves a systematic approach that's the same every time.Here鈥檚 a simple, step-by-step guide you can follow:
  • **Replace**: Start by switching the function notation \(f(x)\) with \(y\). It's easier when you work with an equation.
  • **Swap Variables**: Exchange every instance of \(x\) with \(y\) and \(y\) with \(x\). This gives you the equation you need for finding the inverse.
  • **Solve for \(y\)**: Rearrange the equation to solve for \(y\). This step will vary substantially depending on the original function.
  • **Finalize**: Once \(y\) is isolated, replace it with the inverse function notation \(f^{-1}(x)\).
It鈥檚 important to note that not all functions will have inverses, only those with a one-to-one correspondence between inputs and outputs.
Notation for Inverse Functions
In mathematics, notations simplify the way we express complex ideas and processes. When dealing with inverse functions, the notation \(f^{-1}(x)\) is used to denote the inverse of a function \(f(x)\).Here are some key points about this notation:
  • **Inverse Does Not Mean Reciprocal**: Though it might look like it, the "negative one" superscript is **not** an exponent. It does not imply a reciprocal such as \(\frac{1}{f(x)}\).
  • **Function Behavior**: Using \(f^{-1}(x)\) implies that when you apply \(f\) to \(f^{-1}(x)\), you will get \(x\) back because they cancel each other out like reversing a process.
  • **Using and Recognizing**: When you see \(f^{-1}(x)\), understand it signifies a mirror-like, opposite operation to \(f(x)\).
Grasping this notation is crucial. It reveals a deep and elegant feature of functions鈥攖hat each inverse function acts as a 'counterbalance' to its original function, allowing you to "undo" each step taken by \(f\).

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

Solve each equation. See Example \(8 .\) $$ \log x+\log (x-48)=2 $$

Bacterial Culture. A bacteria culture doubles in size every 24 hours. By how much will it have increased in 36 hours?

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest tenth. See Using Your Calculator: Solving Exponential Equations Graphically or Solving Logarithmic Equations Graphically. $$ \ln (2 x+5)-\ln 3=\ln (x-1) $$

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 2 \log _{3} x-\log _{3}(x-4)=2+\log _{3} 2 $$

Solve each equation. Give the exact solution and an approximation to four decimal places. See Example 5 . $$ e^{2.9 x}=4.5 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.