/*! 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 117 Suppose \(p\) and \(q\) are rati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 117

Suppose \(p\) and \(q\) are rational numbers. Define functions \(f\) and \(g\) by \(f(x)=x^{p}\) and \(g(x)=x^{q} .\) Explain why $$ (f \circ g)(x)=x^{p q} $$

Short Answer

Expert verified
By applying function composition, we have \((f \circ g)(x) = f(g(x))\). Substituting the given functions, we get \(f(g(x)) = f(x^q) = (x^q)^p\). Simplifying the expression using the exponentiation rule, we obtain \((x^q)^p = x^{pq}\). Thus, \((f \circ g)(x)=x^{pq}\).

Step by step solution

01

Understand function composition

Function composition is a way to apply one function to the output of another function. The composition of two functions \(f\) and \(g\) is written as \((f \circ g)(x)\), which means applying function \(f\) on the output of function \(g\). Mathematically, it is represented as \((f \circ g)(x) = f(g(x))\). In this step, we have to apply function \(f\) to the result of function \(g(x)\).
02

Apply the given functions

The given functions are \(f(x) = x^p\) and \(g(x) = x^q\). Now, we need to apply function \(f\) to the output of function \(g\). We have \(g(x) = x^q\). Substituting this into function \(f\), we get: $$ f(g(x))=f(x^q)=(x^q)^p $$
03

Simplify the expression

Now, we need to simplify the expression \((x^q)^p\). Using the exponentiation rule, we get $$ (x^q)^p = x^{pq} $$
04

Conclude

By applying function \(f\) to the output of function \(g\), we obtain the equation: $$ (f \circ g)(x)=x^{pq} $$ Therefore, the composition of the functions \(f(x) = x^p\) and \(g(x) = x^q\) results in the function \((f \circ g)(x) = x^{pq}\).

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!

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

Suppose \(q(x)=2 x^{3}-3 x+1\) (a) Show that the point (2,11) is on the graph of \(q\). (b) Show that the slope of a line containing (2,11) and a point on the graph of \(q\) very close to (2,11) is approximately 21 . [Hint: Use the result of Exercise \(17 .]\)

Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).

Give an example of a polynomial \(p\) of degree 8 such that \(p(2)=3\) and \(p(x) \geq 3\) for all real numbers \(x\).

Explain why the composition of two rational functions is a rational function.

Suppose \(s(x)=\frac{x^{2}+2}{2 x-1}\) (a) Show that the point (1,3) is on the graph of \(s\). (b) Show that the slope of a line containing (1,3) and a point on the graph of \(s\) very close to (1,3) is approximately -4 [Hint: Use the result of Exercise \(25 .]\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.