/*! 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 51 Determine the function that is g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 51

Determine the function that is graphed if the graph of \(f(x)=\sqrt{x}\) is reflected about the \(x\) -axis and then vertically compressed by a factor of \(\frac{1}{3}\)

Short Answer

Expert verified
\( g(x) = -\frac{1}{3} \sqrt{x} \)

Step by step solution

01

Identify the Original Function

The original function given is \( f(x) = \sqrt{x} \).
02

Reflect About the x-axis

Reflecting the function about the \(x\)-axis means multiplying the function by -1. Thus, the reflected function becomes \( -\sqrt{x} \).
03

Apply Vertical Compression

After reflecting, compress the function vertically by multiplying it by \( \frac{1}{3} \). Hence, the vertically compressed function is \( -\frac{1}{3} \sqrt{x} \).
04

Write the Final Function

The final function, after reflecting about the \(x\)-axis and then compressing vertically by a factor of \( \frac{1}{3} \), is \( g(x) = -\frac{1}{3} \sqrt{x} \).

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.

Reflection About x-axis
When a function is reflected about the x-axis, it essentially flips over the x-axis. This type of transformation takes every point of the original function and moves it to the opposite side of the x-axis. The y-values of each point become their negatives.

For example, given the function \(f(x) = \sqrt{x} \), reflecting it about the x-axis gives us \(-f(x) = -\sqrt{x} \).

Visualizing this on a graph, every point \((a, b)\) on \(f(x)\) becomes \((a, -b)\) on \(-f(x)\).

This change mirrors the graph, flipping it upside down while maintaining the same x-values.
Vertical Compression
A vertical compression pushes the graph towards the x-axis, making it 'flatter'. To compress a function vertically by a factor of \(k\), you multiply the function by \(k\), where \(0 < k < 1\).

In our example, after reflecting \(f(x) = \sqrt{x} \) to get \( -\sqrt{x}\), we compress it by a factor of \( 1/3 \). This gives us:

\[ g(x) = -\frac{1}{3}\sqrt{x} \].

The effect of this transformation is that the y-values of the original function are one third of their previous values, making the graph less steep.
Square Root Function
The square root function, \( f(x) = \sqrt{x} \), is a fundamental function in mathematics.
It produces the non-negative square root of x.

  • The domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \), as you cannot take the square root of a negative number and get a real number.
  • The range of \( f(x) = \sqrt{x} \) is also \( y \geq 0 \) because square roots produce non-negative results.

Graphically, \( f(x) = \sqrt{x} \) starts at the origin (0,0) and increases slowly, curving upwards.
As x increases, \( f(x) \) increases but at a decreasing rate.

Understanding the behavior of the square root function is crucial for applying transformations like reflection and vertical compression.

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 Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=x^{4}+x^{3}-3 x^{2}-x+2 $$

Use the Factor Theorem to prove that \(x-c\) is a factor of \(x^{n}-c^{n}\) for any positive integer \(n\).

Solve each equation in the real number system. $$ 3 x^{3}+4 x^{2}-7 x+2=0 $$

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=2 x^{3}-4 x^{2}-10 x+20 $$

If \(f(x)=4 x+3,\) find \(f\left(\frac{x-3}{4}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

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.