/*! 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 71 Let the domain of \(f(x)\) be \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 71

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$3 f\left(\frac{1}{4} x\right)$$

Short Answer

Expert verified
Domain: [-4, 8], Range: [0, 9]

Step by step solution

01

Identify the Transformation Operations

The expression given is \( 3f\left(\frac{1}{4}x\right) \). There are two transformations applied: \( f\left(\frac{1}{4}x\right) \) is a horizontal stretch by a factor of 4, and \( 3f(x) \) is a vertical stretch by a factor of 3.
02

Adjust the Domain for Horizontal Stretch

The original domain is \([-1, 2]\). The function \( f\left(\frac{1}{4}x\right) \) stretches the domain by a factor of 4. To find the new domain, multiply each endpoint of the original domain by 4. Thus, \(-1 \times 4 = -4\) and \(2 \times 4 = 8\), giving us the new domain \([-4, 8]\).
03

Adjust the Range for Vertical Stretch

The original range is \([0, 3]\). The function \( 3f(x) \) stretches the range by a factor of 3. To find the new range, multiply each endpoint of the original range by 3. Thus, \(0 \times 3 = 0\) and \(3 \times 3 = 9\), giving us the new range \([0, 9]\).
04

Combine Transformed Domain and Range

After applying the transformations, the new domain becomes \([-4, 8]\) and the new range becomes \([0, 9]\). This reflects the composition of both transformations on the function.

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.

Understanding Horizontal Stretch
A horizontal stretch affects how a function behaves along the x-axis, widening the graph out horizontally. It alters the way points are distributed over the x-axis by stretching or compressing them. For instance, when you have a function such as \( f\left(\frac{1}{4}x\right) \), it means you will stretch the x-values four times further apart.

Here鈥檚 what you need to remember about horizontal stretches:
  • They change the domain of the function.
  • The factor by which you multiply the x-coordinates is the reciprocal of what appears inside the function with x. So, \(\frac{1}{4}\) becomes 4 as the stretching factor.
  • The original domain end-points of \([-1, 2]\) will be transformed by multiplying them by 4, widening them to \([-4, 8]\).
This means if you imagine the function鈥檚 original graph, it鈥檚 now broadened to cover a larger span on the horizontal axis.
Exploring Vertical Stretch
Vertical stretch involves the y-axis, influencing the output values of a function. It effectively pulls the graph upwards or downwards, amplifying or diminishing the effect. In the vertical stretch case, like \( 3f(x) \), each y-coordinate is three times as far from the x-axis than it was originally.

Key points to note about vertical stretches:
  • They modify the range of the function.
  • Every y-value is multiplied by the stretch factor. In our example, this factor is 3.
  • The original range \([0, 3]\) scales up to \([0, 9]\) when multiplied by 3.
This ensures the function's graph reaches higher peaks and deeper valleys compared to the original.
Domain and Range Transformations
When functions undergo transformations, understanding how domain and range change is crucial. During a horizontal or vertical stretch, these transformations modify where and how the function plots on a graph.

Let鈥檚 break it down:
  • **Domain Transformations**: These are related to horizontal changes. With a horizontal stretch by factor 4, each element of the domain \([-1, 2]\) is magnified, resulting in the new domain \([-4, 8]\).
  • **Range Transformations**: Pertaining to vertical changes, such as multiplying a function by 3, which swells the range from \([0, 3]\) to \([0, 9]\).
By recognizing changes to domain and range, one can accurately graph transformations. Knowing each stretch factor helps identify the adjusted limits, giving a clearer picture of how the function behaves after transformation. It's like expanding a zoomed-in photo; you get more of the surrounding context in view.

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

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) ) for each function \(f\). Simplify completely. $$f(x)=-6 x^{2}-x+4$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=\frac{1}{x+1}, \quad g(x)=5 x$$

Solve each equation graphically. $$|x|+|x-4|=8$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=2-x, \quad g(x)=\frac{1}{x^{2}}$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) ) for each function \(f\). Simplify completely. \(f(x)=\sqrt{x}\) (Hint: Rationalize the numerator.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

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.