/*! 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 81 Find and simplify the difference... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 81

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=-x^{2}+2 x+4$$

Short Answer

Expert verified
The simplified difference quotient of the function \(f(x)=-x^{2}+2x+4\) is \(-2x-h+2\).

Step by step solution

01

Substitute (x+h) into the function

First, replace all instances of \(x\) with \((x+h)\) in the function \(f(x)=-x^{2}+2x+4\). This results in: \(f(x+h)=-((x+h)^{2})+2(x+h)+4\).
02

Simplify the function f(x+h)

Expand and simplify the result from step 1. Using the formula \((a+b)^2=a^2+2ab+b^2\), the function \(f(x+h)\) simplifies to: \(-x^{2}-2hx-h^{2}+2x+2h+4\).
03

Replace f(x+h) and f(x) in the difference quotient

Now, put the expression for \(f(x+h)\) and the given expression for \(f(x)\) back into the difference quotient formula \(\frac{f(x+h)-f(x)}{h}\), which then becomes: \(\frac{-x^{2}-2hx-h^{2}+2x+2h+4-(-x^{2}+2x+4)}{h}\).
04

Simplify the fraction

Simplifying the numerator by removing like terms results in: \(-2xh-h^{2}+2h\). This simplifies further to: \((-2x-h+2)h\). Then divide each term by \(h\), simplifying it to: \(-2x-h+2\).

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

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{x}$$

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+y^{2}=25$$

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{\frac{1}{3}}(x-4)$$

The regular price of a computer is \(x\) dollars. Let \(f(x)=x-400\) and \(g(x)=0.75 x\) a. Describe what the functions \(f\) and \(g\) model in terms of the price of the computer. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the computer. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the computer, \(f^{\circ}\) g or \(g \circ f\) ? Explain.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.