/*! 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 160 In general, composition of funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 160

In general, composition of functions is not commutative. For example, for the functions \(f\) and \(g\) given by \(f(x)=x+1, g(x)=2 x\), we have \(f(g(x))=2 x+1\) and \(g(f(x))=2 x+2\). Now suppose that we have three functions \(f, g, h\). Then there are six possible compositions of the three, given by \(f(g(h(x))), g(h(f(x))), \ldots\). Give an example of three continuous functions that are defined for all real \(x\) and for which exactly five of the six compositions are the same. (Reprinted with the permission of the Canadian Mathematical Society, this problem was originally published in the Mathematical Mayhem section of Crux Mathematicorum with Mathematical Mayhem, vol. 25,1999, p. 293, problem C87.)

Short Answer

Expert verified
Use functions: \( f(x) = x \), \( g(x) = -x \), and \( h(x) = x \). Five compositions result in \(-x\), one in \(x\).

Step by step solution

01

- Define the simple functions

Start by defining three simple continuous functions for all real numbers. Consider the following functions: 1. \( f(x) = x \) 2. \( g(x) = -x \) 3. \( h(x) = x \)
02

- Compute the compositions

Calculate each of the six possible compositions with the defined functions:1. \( f(g(h(x))) \)2. \( f(h(g(x))) \)3. \( g(f(h(x))) \)4. \( g(h(f(x))) \)5. \( h(f(g(x))) \)6. \( h(g(f(x))) \)
03

- Simplify the compositions

Substitute and simplify each composition.1. \( f(g(h(x))) = f(g(x)) = f(-x) = -x \)2. \( f(h(g(x))) = f(x) = x \)3. \( g(f(h(x))) = g(x) = -x \)4. \( g(h(f(x))) = g(x) = -x \)5. \( h(f(g(x))) = h(-x) = -x \)6. \( h(g(f(x))) = h(-x) = -x \)
04

- Verify the five identical compositions

Notice that five of the six compositions result in \(-x\). These are:1. \( f(g(h(x))) = -x \)2. \( g(f(h(x))) = -x \)3. \( g(h(f(x))) = -x \)4. \( h(f(g(x))) = -x \)5. \( h(g(f(x))) = -x \)
05

- Identify the different composition

The different composition is:\( f(h(g(x))) = x \)

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.

Composition of Functions
The composition of functions is the process of combining two functions to form a new function. It is denoted as (f o g)(x), which means f(g(x)), where g(x) is computed first, and then f is applied to that result. When composing functions, always start with the innermost function and work outwards. For example, in the given problem, to compute f(g(h(x))), we:
  • Apply h(x), which is just x

  • Then apply g to the result of h(x), giving us g(x) = -x

  • Finally, apply f to g(x), resulting in f(-x) = -x

In general, when combining three or more functions, there are multiple possible ways to compose them, which may yield different results. The original problem highlights this principle: out of six possible compositions of f, g, and h, five compositions turn out to be the same (-x), while one differs (x). This illustrates the non-commutativity of function composition.

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

Note that if the edges of a regular octahedron have length 1 , then the distance between any two of its vertices is either 1 or \(\sqrt{2}\). Are there other configurations of six points in \(\mathbb{R}^3\) for which the distance between any two of the points is either 1 or \(\sqrt{2}\) ? If so, find them.

There is no analog of the quadratic formula that solves polynomial equations of degree 5 and higher, such as \(x^5-5 x^4+8 x^3-6 x^2+3 x+3=0\). However, this particular polynomial has two roots that sum to 2 . Using this information, find all solutions.

Suppose we are given an \(m\)-gon (polygon with \(m\) sides, and including the interior for our purposes) and an \(n\)-gon in the plane. Consider their intersection; assume this intersection is itself a polygon (other possibilities would include the intersection being empty or consisting of a line segment). a. If the \(m\)-gon and the \(n\)-gon are convex, what is the maximal number of sides their intersection can have? b. Is the result from (a) still correct if only one of the polygons is assumed to be convex? (Note: A subset of the plane is convex if for every two points of the subset, every point of the line segment between them is also in the subset. In particular, a polygon is convex if each of its interior angles is less than \(\left.180^{\circ}.\right)\)

Note that the three positive integers \(1,24,120\) have the property that the sum of any two of them is a different perfect square. Do there exist four positive integers such that the sum of any two of them is a perfect square and such that the six squares found in this way are all different? If so, exhibit four such positive integers; if not, show why this cannot be done.

If \(A=(0,-10)\) and \(B=(2,0)\), find the point(s) \(C\) on the parabola \(y=x^{2}\) which minimizes the area of triangle \(A B C\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.