/*! 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 20 Determine functions \(g\) and \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

Determine functions \(g\) and \(h\) so that \(f(x)=g(h(x))\). $$f(x)=\sqrt[3]{x-9}$$

Short Answer

Expert verified
The functions are \(g(u) = \sqrt[3]{u}\) and \(h(x) = x-9\).

Step by step solution

01

Identify the Outer Function

First, we need to see the structure of the function \(f(x) = \sqrt[3]{x-9}\). We can observe that the outer function is taking the cube root. Therefore, the outer function \(g(u)\) can be defined as \(g(u) = \sqrt[3]{u}\).
02

Identify the Inner Function

Next, we identify the expression that is inside the cube root. In the function \(f(x) = \sqrt[3]{x-9}\), we notice that the cube root operates on \(x-9\). Therefore, the inner function \(h(x)\) can be defined as \(h(x) = x-9\).
03

Verify the Composite Function

To ensure that our functions are correct, we substitute \(h(x) = x-9\) into \(g(h(x)) = g(u)\) to check if we get back \(f(x)\). Calculate \(g(h(x)) = g(x-9) = \sqrt[3]{x-9}\). This matches our original function \(f(x)\), confirming that \(g(u) = \sqrt[3]{u}\) and \(h(x) = x-9\) are correct.

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 Decomposition
Function decomposition is a strategy in mathematics where a complex function is broken down into simpler, more manageable parts. By doing this, it becomes easier to analyze and work with functions.
In our given example, the function is designed as a composition of two functions, namely the inner function and the outer function.
  • The outer function processes what the inner function outputs.
  • The inner function is contained within and is evaluated first.
Breaking down functions into these parts helps us understand how inputs are transformed step-by-step. Think of function decomposition like peeling an onion; you remove one layer at a time to reveal the core. This layered understanding is fundamental in evaluating composite functions.
Cube Root Function
The cube root function is an essential mathematical operation that deals with finding a number which, when multiplied by itself three times, gives the original number back. When you see the expression \( \sqrt[3]{x} \), it represents the cube root of \( x \). The cube root is similar to the square root, but instead of squaring, you're working with cubes.
  • The cube root of \( 8 \) is \( 2 \) because \( 2 \times 2 \times 2 = 8 \).
  • Unlike square roots, every real number has a real cube root.
  • This includes negative numbers, because the cube of a negative number is also negative.
In our specific function, the cube root function \( g(u) = \sqrt[3]{u} \) processes the result of the inner function \( h(x) \), bringing us to the complete function.
Inner and Outer Functions
A composite function consists of two functions, an inner function and an outer function. This concept is crucial to understand how functions interact in such a composition.
In our scenario:
  • The **inner function** is \( h(x) = x - 9 \). It is the first step to modify the input, preparing it for the next function.
  • The **outer function** is \( g(u) = \sqrt[3]{u} \). It processes the result of the inner function.
Think of this like putting a batch of cookies into an oven. The cookie dough (inner function) is shaped and prepared before being baked by the oven (outer function).
Understanding the roles of inner and outer functions lets us rearrange or simplify complex expressions easily. It's important to remember that the inner function works first within the argument of the outer function. This layered processing exemplifies the beautiful harmony in composition functions.

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

Specify a sequence of transformations to perform on the graph of \(y=x^{2}\) to obtain the graph of the given function. $$f: x \mapsto[3(x-1)]^{2}-6$$

Find the inverse function \(f^{-1}\) and state its domain. $$f(x)=\frac{1}{x+2}$$

Find the domain of the function. $$h(t)=\sqrt[3]{t}$$

Use the functions \(g(x)=x+3\) and \(h(x)=2 x-4\) to find the indicated value or the indicated function. $$\left(g^{-1} \circ h^{-1}\right)(5)$$

Find the domain of the function. $$\\{(-6.2,-7),(-1.5,-2),(0.7,0),(3.2,3),(3.8,3)\\}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.