/*! 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 17 \(h(x)=-\sqrt[4]{x}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 17

\(h(x)=-\sqrt[4]{x}\)

Short Answer

Expert verified
The function \(h(x)=-\sqrt[4]{x}\) represents the negative 4th root of \(x\). The domain of this function is \(x \geq 0\) in the real number system. The graph of this function starts at the origin and gradual declines, leveling off as \(x\) increases.

Step by step solution

01

Review the function

The given function, \(h(x)=-\sqrt[4]{x}\), represents the negative 4th root of \(x\). This indicates that for any input \(x\), \(h(x)\) is equal to the negative value of its 4th root.
02

Determine the domain

For this function, the root operation only allows us to have non-negative numbers as its 'input' values i.e. \(x\) should be greater than or equal to 0. Hence, the domain of this function is \(x \geq 0\). It's important to note that while negative values of \(x\) can be plugged into a 4th root in the real number system (because the result is still a real number), the function here has specifically been defined as the negative 4th root of \(x\), meaning \(x\) should be non-negative.
03

Sketch the graph

To sketch the graph of this function, note that it behaves like the 4th root of \(x\), but with a negative sign. This negative sign flips the graph over the x-axis. So instead of the curve increasing as \(x\) increases, as happens with the positive 4th root of \(x\), the curve will decrease as \(x\) increases. The graph starts at the origin and goes down towards the negative y-axis, gradually leveling off as \(x\) gets larger and larger.

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.

Domain of Functions
Understanding the domain of a function is key to grasping how it behaves. For the function \(h(x) = -\sqrt[4]{x}\), determining the domain involves focusing on which values of \(x\) are permissible.

Since \(h(x)\) is the negative 4th root of \(x\), this requirement dictates that \(x\) must be greater than or equal to zero. Here鈥檚 why: the 4th root is only properly defined for non-negative numbers in real numbers, meaning \(x\) should be zero or positive.
  • For \(x = 0\), we have \(h(0) = 0\), which is valid.
  • For \(x > 0\), \(h(x)\) produces a real, negative result.
  • For \(x < 0\), \(h(x)\) would result in complex outcomes, which aren鈥檛 handled traditionally in real-valued functions like this one.
Thus, the domain of this function is \(x \geq 0\), ensuring that all outputs remain real numbers.
Graphing Functions
Graphing functions involves drawing a visual representation of how a function behaves over its domain. To graph \(h(x) = -\sqrt[4]{x}\), it helps to understand its basic shape and behavior.

The function describes a transformation of the basic 4th root function \(\sqrt[4]{x}\) but flipped over the x-axis, due to the negative sign. Here are the steps:
  • The graph starts at the point (0,0), since the 4th root of 0 is 0, and multiplying by the negative sign keeps it at 0.
  • As \(x\) increases, the value of \(h(x)\) gets more negative (lower), indicating that the graph slopes downward.
  • This downward slope gradually levels off as \(x\) continues increasing, giving the curve a gentle, smooth descent.
With these points, you sketch a curve that begins at the origin and extends into the negative y-region, simulating a gentle bowl opening to the right and downward.
Transformation of Functions
Transformations in functions help us understand changes in the basic shape or position of a graph. For \(h(x) = -\sqrt[4]{x}\), the main transformation to note is the reflection over the x-axis.

Such transformations occur when a function is manipulated, like through multiplication by negative numbers or constants. Let's explore this:
  • The basic function \(\sqrt[4]{x}\) would typically form a curve that arches upward, but the negative sign "flips" this upwards arch downwards.
  • Rather than rising as \(x\) grows, this function falls, producing a mirror image of what you鈥檇 expect from \(\sqrt[4]{x}\).
  • This flip results in a curve that drifts steadily downward, rather than ascending, as \(x\) increases.
By grasping this transformation, you gain insights into how changing signs and numbers in function definitions impact their graphical layout.

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 domain of the function \(y=a \sqrt{x}\) is \(x \geq 0\)

In Exercises 35鈥�46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=x^4+2 $$

\(f(x)=(6 x)^{1 / 2}+3\)

Do functions of the form \(y=x^{m / n}\), where \(m\) and \(n\) are positive integers, have inverse functions? Justify your answer with examples.

\(f(x)=\sqrt[3]{x}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.