/*! 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 120 Think About It, determine (if po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 120

Think About It, determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\). $$g(x)=f(2 x)$$

Short Answer

Expert verified
The zeros of the function \(g(x) = f(2x)\) are at \(x = r_{1}/2\), \(x = r_{2}/2\), and \(x = r_{3}/2\).

Step by step solution

01

Identify the zeros of \(f(x)\)

According to the problem, the zeros of \(f(x)\) are at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\). These are the x-values for which the function \(f(x)\) reduces to zero.
02

Apply the transformation to the zeros

The function \(g(x)\) is defined in terms of \(f(x)\) as \(g(x) = f(2x)\). The factor of 2 in the argument of \(f\) indicates that the x-variable in \(f\) is stretched by a factor of 1/2. Consequently, the zeros for the function \(g(x)\) will be the zeros of \(f(x)\) scaled by 1/2. Thus, \(r_{1}/2\), \(r_{2}/2\), and \(r_{3}/2\).
03

Determine the zeros of \(g(x)\)

From step 2, the zeros of \(g(x) = f(2x)\) are at \(x = r_{1}/2\), \(x = r_{2}/2\), and \(x = r_{3}/2\).

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.

Function Transformations
A function transformation involves modifying a function in a way that affects its graph. In the exercise, we're dealing with a specific type of transformation applied to the function \(f(x)\), resulting in a new function \(g(x) = f(2x)\). This involves a horizontal stretch or compression.
  • A horizontal transformation changes the function along the x-axis.
  • When multiplying the input \(x\) by a factor, it causes the graph of the function to stretch or compress horizontally.
In this case, \(f(2x)\) implies a horizontal compression by a factor of 1/2. Each x-coordinate of the zero from \(f(x)\) is halved. This means that the zeros of \(f(x)\), which are \( r_1, r_2, \) and \( r_3 \), become \( r_1/2, r_2/2, \) and \( r_3/2 \) in \(g(x)\). Understanding this transformation is key to identifying how functions behave when their equations are altered.
Precalculus
Precalculus serves as a bridge between algebra and calculus, focusing on concepts that are fundamental to understanding calculus. In this exercise, we are particularly concerned with understanding how transformations affect functions. These ideas are critical in precalculus as they provide the groundwork for understanding more complex concepts in calculus.
  • Precalculus helps in understanding the properties of functions, including zeros, intercepts, and transformations.
  • It prepares students for calculus by introducing the concept of limiting values and rate changes.
  • The manipulation of functions, such as the transformation seen in \(g(x) = f(2x)\), helps in visualizing shifts, stretches, and compressions of graphs.
Understanding function transformations like these ensures a robust grasp of the behavior of different types of functions, which is a cornerstone concept in precalculus.
Scale Factors
Scale factors are crucial in understanding how functions are transformed. They dictate the degree to which a function is expanded or compressed and play a significant role in the general study of functions.In the given transformation \(g(x) = f(2x)\), a scale factor of 2 is used to multiply the x-variable. This scale factor indicates a horizontal compression of the function.
  • A positive scale factor greater than 1 compresses the graph horizontally.
  • A positive scale factor between 0 and 1 stretches the graph horizontally.
  • The zeros of the function are directly affected by this scaling.
Thus, the transformation results in each x-coordinate of the zero being "scaled down" by a factor of 1/2. In general, recognizing how scale factors affect the graph of a function is critical for understanding graphical transformations.

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

Depreciation A laptop computer that costs \(\$ 1150\) new has a book value of \(\$ 550\) after 2 years. (a) Find the linear model \(V=m t+b .\) (b) Find the exponential model \(V=a e^{k t} .\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

Finding the Rational Zeros of a Polynomial, find the rational zeros of the polynomial function. $$f(z)=z^{3}+\frac{11}{6} z^{2}-\frac{1}{2} z-\frac{1}{3}=\frac{1}{6}\left(6 z^{3}+11 z^{2}-3 z-2\right)$$

In Exercises 73 and \(74,\) use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where s represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

Think About It, determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\). $$g(x)=-f(x)$$

Learning Curve The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked t days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k )\) (b) How many days should pass before this employee is producing 25 units per day?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure 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.