/*! 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 62 Solve. Suppose that \(F\) is a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 62

Solve. Suppose that \(F\) is a one-to-one function and that \(F\left(\frac{1}{2}\right)=-0.7\) a. Write the corresponding ordered pair. b. Name one ordered pair that we know is a solution of the inverse of \(F,\) or \(F^{-1}\)

Short Answer

Expert verified
(\frac{1}{2}, -0.7) is for F; (-0.7, \frac{1}{2}) is for F^{-1}.

Step by step solution

01

Identify F as a function

Given that the function value is \( F\left(\frac{1}{2}\right) = -0.7 \), we interpret this as the function \( F \) mapping the number \( \frac{1}{2} \) to \( -0.7 \).
02

Write the corresponding ordered pair for F

Since \( F\left(\frac{1}{2}\right) = -0.7 \), the ordered pair for the function \( F \) is \( \left( \frac{1}{2}, -0.7 \right) \).
03

Understand the concept of an inverse function

If \( F \) is a one-to-one function, then its inverse, \( F^{-1} \), reverses the roles of input and output, meaning if \( F(a) = b \), then \( F^{-1}(b) = a \).
04

Determine the ordered pair for the inverse function

The ordered pair \( \left( \frac{1}{2}, -0.7 \right) \) for function \( F \) translates to \( (-0.7, \frac{1}{2}) \) for the inverse function \( F^{-1} \). This is because \( F^{-1}(-0.7) = \frac{1}{2} \).

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
In mathematics, a **one-to-one function**, also known as an injective function, is a type of function where every element in the domain is mapped to a distinct and unique element in the range. This means no two different inputs can have the same output.
  • If each input has a unique output, the function can be reversed or inverted.
  • A one-to-one function passes the horizontal line test鈥攁ny horizontal line passes through the graph of the function at most once.
  • This property is crucial for the existence of an inverse function.
For example, if a function transforms 2 into 5 and only 2 into 5, then that's a sign of a one-to-one function.
Ordered pairs
An **ordered pair** is a mathematical concept used to describe the input and output relationship in a function in the form \( (x, y) \).
  • The first element, \( x \), is known as the input or independent variable.
  • The second element, \( y \), is known as the output or dependent variable.
Ordered pairs are a simple way to express the idea that each input \( x \) leads to one output \( y \). For instance, if a function \( F \) maps \( x = \frac{1}{2} \) to \( y = -0.7 \), then the ordered pair is \( \left( \frac{1}{2}, -0.7 \right) \). This indicates the precise relationship given by the function at that specific point.
Function mapping
**Function mapping** refers to the process by which a function assigns, or "maps," each element in the domain to an element in the range. This idea of mapping is central to understanding how functions work.
  • A function can be thought of as a machine that takes an input (domain element) and produces an output (range element).
  • For our given function \( F \), mapping is illustrated as \( F \left( \frac{1}{2} \right) = -0.7 \).
  • This means that when the function \( F \) "receives" \( \frac{1}{2} \), it "outputs" \( -0.7 \).
Thus, understanding function mappings provides clarity on how inputs are transformed into outputs, which is a foundational concept in function analysis and interpretation.
Inverse relationship
The concept of an **inverse relationship** in functions is fascinating and crucial, especially in the context of one-to-one functions. An inverse function essentially "undoes" the operation of the original function. If a function \( F \) maps \( x \) to \( y \), an inverse function \( F^{-1} \) would map \( y \) back to \( x \).
  • In practical terms, if \( F(a) = b \), then \( F^{-1}(b) = a \).
  • This reversal of roles highlights the unique one-to-one relationship between input and output in the function.
  • For our example, given \( F \left( \frac{1}{2} \right) = -0.7 \), the inverse function \( F^{-1} \) will map \( -0.7 \) back to \( \frac{1}{2} \).
Thus, the ordered pair \( \left( -0.7, \frac{1}{2} \right) \) represents the inverse relationship and shows how the roles are reversed in he inverse function.

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

Use the formula \(A=P e^{r t}\) to solve. See Example 8. Find the amount of money for which a 2500 dollars certificate of deposit is redeemable if it has been paying \(10 \%\) interest compounded continuously for 3 years.

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x+4} $$

Solve each equation for \(x\). Give an exact solution and a four-decimal-place approximation. See Examples 3 and 7. $$ \log (2 x+1)=-0.5 $$

Graph \(y=2^{x}\) and \(y=\left(\frac{1}{2}\right)^{-x}\) on the same set of axes Describe what you see and why.

Solve each equation for \(x .\) See Example 4. \(2^{x}=64\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.