/*! 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 9 For the following exercises, fin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 9

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=2-x\)

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = 2 - x \).

Step by step solution

01

Replace f(x) with y

Start with the given function: \( f(x) = 2 - x \). Replace \( f(x) \) with \( y \) to set up the equation: \( y = 2 - x \). This substitution will help us find the inverse function by solving for \( x \) in terms of \( y \).
02

Solve for x in terms of y

We need to express \( x \) in terms of \( y \). Start with the equation \( y = 2 - x \). Add \( x \) to both sides to get \( x + y = 2 \). Then, subtract \( y \) from both sides to solve for \( x \): \( x = 2 - y \).
03

Replace x with f^{-1}(x) and y with x

To express the inverse function, swap \( x \) and \( y \), changing \( y \) to \( x \) and \( x \) to \( f^{-1}(x) \). This gives us \( f^{-1}(x) = 2 - x \).

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 Notation
When dealing with functions, it is common to encounter different ways that functions are represented, known as function notation. The notation \( f(x) \) indicates that \( f \) is a function of \( x \), the input variable. In simple terms, it is like a formula or a machine where you put in a value for \( x \) and get out a result.
  • The expression \( f(x) = 2 - x \) tells us that for any input \( x \), to find the output, we subtract \( x \) from 2.
  • Inverse functions are often represented with \( f^{-1}(x) \) notation. This means that this new function undoes the work of the original function.
This notation helps us clearly see the relationship between inputs and outputs and also allows us to communicate the idea of reversing a function with the inverse.
Solving Equations
Solving equations is a crucial step in finding inverse functions, as you need to rearrange the function equation to express the input variable in terms of the output variable. For the function \( f(x) = 2 - x \), we start by letting \( y = f(x) \), which translates to \( y = 2 - x \). Our goal is to manipulate this equation to solve for \( x \) as a function of \( y \).
  • First, to isolate \( x \), we add \( x \) to both sides: \( x + y = 2 \).
  • Then, subtract \( y \) from both sides, which gives us \( x = 2 - y \).
Once you solve for \( x \), you can appropriately swap \( x \) and \( y \) to express the inverse function in standard form.
Inverse Operations
Inverse operations are key in finding the inverse of a function. The concept of inverse operations is about using operations that reverse or "undo" each other. In the original function \( f(x) = 2 - x \), we subtract \( x \) from 2 to find the output.
  • The inverse function, \( f^{-1}(x) \), must undo this subtraction by using addition, retracing steps to the original input.
  • When we set up \( y = 2 - x \) and solve for \( x \), we reverse the operations to express \( x \) in terms of \( y \), leading to \( x = 2 - y \).
Finally, to find the inverse function, we switch the variables, yielding \( f^{-1}(x) = 2 - x \), reinforcing how inverse operations effectively reverse the function's process.

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

Describe all numbers \(x\) that are at a distance of \(\frac{1}{2}\) from the number -4 . Express this set of numbers using absolute value notation.

For the following exercises, graph each function using a graphing utility. Specify the viewing window. \(f(x)=-0.1|0.1(0.2-x)|+0.3\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-3|x-2|-1\)

For the following exercises, find a domain on which each function \(f\) is one- to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain. \(f(x)=(x-6)^{2}\)

For the following exercises, determine whether the function is odd, even, or neither. \(h(x)=\frac{1}{x}+3 x\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure 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.