/*! 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 22 Decide whether each function is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 22

Decide whether each function is one-to-one. Do not use a calculator. $$y=\frac{1}{x+2}$$

Short Answer

Expert verified
The function \( y = \frac{1}{x+2} \) is one-to-one.

Step by step solution

01

Definition of One-to-One Function

A function is considered one-to-one if every output value (y-value) is mapped from exactly one input value (x-value). In other words, no horizontal line should intersect the graph of the function at more than one point.
02

Analyze the Given Function

The function provided is \( y = \frac{1}{x+2} \). This is a rational function that can be rewritten in terms of its transformation: it's essentially the reciprocal function \( \frac{1}{x} \), shifted 2 units to the left.
03

Applying the Horizontal Line Test

To determine if the function is one-to-one, we apply the horizontal line test. A horizontal line will intersect the graph of \( y = \frac{1}{x+2} \) only once for any y-value, except at points where the function is undefined (such as \( x = -2 \)).
04

Conclusion

Since no horizontal line intersects the graph at more than one point, the function \( y = \frac{1}{x+2} \) is one-to-one.

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.

Horizontal Line Test
To determine if a function is one-to-one, we apply the horizontal line test. This is a straightforward test. You imagine drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph only once, the function is one-to-one. This ensures each output is linked to a unique input.

In the provided exercise, we're checking the function \( y = \frac{1}{x+2} \). If you picture its graph, you'll notice that no horizontal line crosses it more than once, except where the function is undefined (like \( x = -2 \)). That means the function passes the horizontal line test! Therefore, we confirm that it is one-to-one.
Rational Function
A rational function is a type of function defined by the quotient of two polynomials. They generally take the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

The function \( y = \frac{1}{x+2} \) is a simple rational function where \( P(x) = 1 \) and \( Q(x) = x + 2 \). Rational functions often have specific characteristics, like vertical asymptotes and horizontal or oblique asymptotes.

For \( y = \frac{1}{x+2} \), there's a vertical asymptote at \( x = -2 \) because the denominator becomes zero here, making the function undefined. Understanding these traits helps identify the function's behavior across its domain.
Function Transformation
Function transformation involves shifting, stretching, compressing, or reflecting the graph of a function. By understanding these transformations, we can easily recognize how basic functions are modified.

For our example, \( y = \frac{1}{x+2} \) can be viewed as a transformed version of the basic reciprocal function \( y = \frac{1}{x} \). Here's how:
  • Horizontal Shift: Adding 2 within the denominator \( (x + 2) \) shifts the graph 2 units to the left. This transformation changes the vertical asymptote from \( x = 0 \) to \( x = -2 \).
These transformations don't alter the one-to-one nature of the function but help in visualizing how the graph fits onto the coordinate plane. Understanding transformations is key in recognizing and sketching the graphs of various functions quickly.

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

The interest rate stated by a financial institution is sometimes called the nominal rate. If interest is compounded, the actual rate is, in general, higher than the nominal rate, and is called the effective rate. If \(r\) is the nominal rate and \(n\) is the number of times interest is compounded annually, then $$R=\left(1+\frac{r}{n}\right)^{n}-1$$ is the effective rate. Here, \(R\) represents the annual rate that the investment would earn if simple interest were paid. Find the effective rate to the nearest hundredth of a percent if the nominal rate is \(3 \%\) and interest is compounded quarterly.

Account Estimate how long it will take for \(\$ 5000\) to grow to \(\$ 8400\) at an interest rate of \(6 \%\) if interest is compounded (a) semiannually; (b) continuously.

surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$ f(x)=31.5+1.1 \log (x+1) $$ models salinity to depths of 1000 meters at a latitude of \(57.5^{\circ} \mathrm{N} .\) The variable \(x\) is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D\), Global Physical Climatology, Academic Press.) Estimate the salinity at a depth of 500 meters.

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\frac{4}{3} \ln m-\frac{2}{3} \ln 8 n-\ln m^{3} n^{2}$$

The earthquake off the coast of Northern Sumatra on Dec. \(26,2004,\) had a Richter scale rating of 8.9 (a) Express the intensity of this earthquake in terms of \(I_{0}\). (b) Aftershocks from this quake had a Richter scale rating of \(6.0 .\) Express the intensity of these in terms of \(I_{0}\) (c) Compare the intensities of the 8.9 earthquake to the 6.0 aftershock.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.