/*! 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 6 Assume that the function \(f\) i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Assume that the function \(f\) is a one-to-one function. If \(f(1)=4,\) find \((f(1))^{-1}\)

Short Answer

Expert verified
\((f(1))^{-1} = \frac{1}{4}\).

Step by step solution

01

Identify what is being asked

We are asked to find \((f(1))^{-1}\). This refers to the inverse of the function value \(f(1)\). Since \(f(1) = 4\), \((f(1))^{-1}\) would be the reciprocal of 4.
02

Calculate the Reciprocal

The reciprocal of a number is 1 divided by that number. For the function value \(f(1) = 4\), the reciprocal is \(\frac{1}{4}\).

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 Function
A one-to-one function is a special type of function where each output is only associated with one unique input. This means no two different inputs can have the same output.
This characteristic is crucial when finding inverse functions because it ensures that each output has a unique inverse input. Without this property, finding an inverse would be ambiguous.
Some characteristics of one-to-one functions include:
  • If the function is graphed, it will pass the "horizontal line test," which means any horizontal line will intersect the function's graph at most once.
  • A formal way to check if a function is one-to-one is to see if the inverse function exists, meaning there is a specific rule or function that "undoes" the original.
  • One-to-one functions can be increasing or decreasing, but they won't switch directions, avoiding any repetition of output values.
Reciprocal
The reciprocal is a mathematical concept that involves flipping a number. Specifically, if you have a number, its reciprocal is 1 divided by that number.
For example, the reciprocal of 4 is \(\frac{1}{4}\). This operation is significant in various mathematical processes where you need to "undo" a multiplication by that number.
  • Multiplying a number by its reciprocal gives you 1, essentially cancelling each other out.
  • Reciprocals are commonly used in solving equations and simplifying expressions.
  • Reciprocal operation is straightforward with non-zero numbers. However, note that zero doesn't have a reciprocal due to division by zero being undefined.
Function Value
The function value is the output you get when you substitute an input into a function. It's written in the format \( f(x) = y \), where \( x \) is the input and \( y \) is the output.
In our original exercise, the function value \( f(1) \) is equal to 4. This means when inputting 1 into the function \( f \), the output is 4.
Some key points about function values include:
  • They are often found by plugging the input into the function's equation and solving.
  • Function values are unique for each input in one-to-one functions, preventing repetition of outputs.
  • Finding a function value helps in understanding the behavior and trend of the function over its domain.
Inverse Operation
Inverse operations are operations that "undo" each other. These are used to reverse the effect of the original operation and return to the initial value.
For example, addition and subtraction are inverse operations, as are multiplication and division.
  • When dealing with functions, the inverse function "reverses" the original function such that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
  • Finding an inverse operation often involves rearranging or reformulating the original operation or function.
  • Being able to identify and apply inverse operations is crucial in solving equations, transforming functions, and mathematical problem-solving in general.

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

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\) $$ h(x)=4+\sqrt[3]{x} $$

Given each function, evaluate: \(f(-1), f(0), f(2), f(4)\). $$ f(x)=\left\\{\begin{array}{lll} 7 x+3 & \text { if } & x<0 \\ 7 x+6 & \text { if } & x \geq 0 \end{array}\right. $$

Describe how each function is a transformation of the original function \(f(x)\). $$ f(2 x) $$

Determine the interval(s) on which the function is concave up and concave down. $$ b(x)=\sqrt[3]{-x-6} $$

For each function below, find \(f^{-1}(x)\). $$ f(x)=3-x $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.