/*! 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 47 Express the function in the form... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 47

Express the function in the form \(f \circ g \circ h\) \(H(x)=\sqrt[8]{2+|x|}\)

Short Answer

Expert verified
\(H(x) = f(g(h(x)))\) with \(h(x)=|x|\), \(g(x)=2+x\), \(f(x)=\sqrt[8]{x}\).

Step by step solution

01

Define the innermost function

Let's start by defining the innermost function associated with the absolute value. In this case, we have that the expression inside the function is \\( h(x) = |x| \)
02

Define the middle function

The result from Step 1, \( h(x) \), is used to define the next part of the function. We add 2 to the absolute value to create this new function: \\( g(x) = 2 + x \)
03

Define the outermost function

The result from Step 2 serves as input to the final function, where we take the eighth root of the previous results: \\( f(x) = \sqrt[8]{x} \)
04

Combine the functions

Combine the functions in the correct order such that\\( (f \circ g \circ h)(x) = f(g(h(x))) \) which translates to: \\( f(g(h(x))) = \sqrt[8]{2 + |x|} \).\Thus, we have expressed \( H(x) \) as \( f \circ g \circ h \).

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.

Innermost Function
The first step in function composition is identifying the innermost function. This is the function you apply first. In our exercise, the expression inside the outermost part is \( |x| \). Here, the absolute value function \( h(x) = |x| \) is the innermost function.
  • The absolute value function is crucial when you want to ensure the number remains non-negative.
  • It transforms any negative input to its positive counterpart.
So when you start with an input \( x \), applying \( h(x) \) returns the value without its sign. It's like the foundation of the function composition where everything begins.
Middle Function
Next in the lineup is the middle function. Once you've dealt with the innermost function, the result becomes an input for the middle function.In our scenario, once the absolute value \( |x| \) is obtained as \( h(x) \), we proceed to the middle function \( g(x) \). This is defined by \( g(x) = 2 + x \), which requires adding 2 to the result of \( h(x) \).
  • This step makes sure that every processed value is shifted by 2 units.
  • It essentially adjusts the entire function by a fixed amount before the next transformation.
This middle step adds a layer of transformation that prepares the values for the outermost function.
Outermost Function
Finally, we reach the outermost function. This is the last transformation applied in the function composition. Once the middle function \( g(x) \) is computed, the result flows into the outermost function.In our exercise, the outermost function is \( f(x) = \sqrt[8]{x} \). This takes the previous result, \( 2 + |x| \), and finds its eighth root.
  • The eighth root operation is a more complex transformation that compresses values towards 1, especially if they are greater than 1.
  • It's the step where the outcome becomes more intricate, combining all previous transformations into the final form of the function.
Through this systematic composition, you seamlessly transition through the innermost to the outermost function, thereby creating an elegant full piece in the function \( H(x) \).

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

Express the function in the form \(f \circ g\) \(F(x)=\left(2 x+x^{2}\right)^{4}\)

Solve each equation for \(x\) . $$(a)2^{x-5}=3 \quad \text { (b) } \ln x+\ln (x-1)=1$$

(a) Suppose \(f\) is a one-to-one function with domain \(A\) and range \(B .\) How is the inverse function \(f^{-1}\) defined? What is the domain of \(f^{-1} ?\) What is the range of \(f^{-1} ?\) (b) If you are given a formula for \(f,\) how do you find a formula for \(f^{-1} ?\) (c) If you are given the graph of \(f,\) how do you find the graph of \(f^{-1} ?\)

If you graph the function $$f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}$$ you'll see that \(f\) appears to be an odd function. Prove it.

Data points \((x, y)\) are given. (a) Draw a scatter plot of the data points. (b) Make semilog and log-log plots of the data. (c) Is a linear, power, or exponential function appropriate for modeling these data? (d) Find an appropriate model for the data and then graph the model together with a scatter plot of the data. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {2} & {4} & {6} & {8} & {10} & {12} \\\ \hline y & {0.08} & {0.12} & {0.18} & {0.26} & {0.35} & {0.53} \\\ \hline\end{array}$$ \(\begin{array}{|c|c|c|c|c|c|}\hline x & {5} & {10} & {15} & {20} & {25} & {30} \\\ \hline y & {0.013} & {0.046} & {0.208} & {0.930} & {4.131} & {18.002} \\\ \hline\end{array}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete 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.