/*! 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 61 $$ \text { For each function, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 61

$$ \text { For each function, find and simplify } f(x+h) \text { . } $$ $$ f(x)=2 x^{2}-5 x+1 $$

Short Answer

Expert verified
\( f(x+h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \).

Step by step solution

01

Understand the Problem

We are given a function \( f(x) = 2x^2 - 5x + 1 \). The task is to find \( f(x+h) \) where \( h \) is another variable, and then simplify the expression.
02

Substitute \( x+h \) into the function

Replace every instance of \( x \) in the function \( f(x) \) with \( x + h \). This yields:\[ f(x+h) = 2(x+h)^2 - 5(x+h) + 1 \].
03

Expand \( (x+h)^2 \)

Compute \( (x+h)^2 \) using the binomial expansion:\[ (x+h)^2 = x^2 + 2xh + h^2 \].Substitute this back into the function:\[ f(x+h) = 2(x^2 + 2xh + h^2) - 5(x+h) + 1 \].
04

Distribute and Simplify Terms

Distribute the 2 into the expanded quadratic and simplify:\[ f(x+h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \].Combine any like terms if possible: there are none to combine here.
05

Final Function Check

Double-check your expansion and distribution for accuracy. Verify that the expression corresponds to the steps taken and is fully simplified.

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 Substitution
Function substitution is a key concept in understanding how functions work, especially when variables are introduced or adjusted. In this exercise, we're working with a function, \( f(x) = 2x^2 - 5x + 1 \). Our task is to replace every instance of \( x \) in the function with \( x+h \).

This substitution transforms the function into \( f(x+h) \). It is like finding the function's value at a point shifted by \( h \). It helps us determine how the output of the function changes as the input changes by \( h \).
  • Replace each \( x \) with \( x+h \) in the function's expression.
  • In our example: \( f(x+h) = 2(x+h)^2 - 5(x+h) + 1 \).
This new expression is not simplified yet, but the substitution gives us a starting point.
Polynomial Expansion
Once the substitution is done, we move to polynomial expansion, an essential step when dealing with expressions like \( (x+h)^2 \). Here, we expand \( (x+h)^2 \) using the binomial theorem.

The binomial theorem tells us how to expand powers of sums. So, \((x+h)^2 = x^2 + 2xh + h^2 \). The expansion explains how each term in \( x+h \) interacts with itself when squared.
  • Start with \((x+h)^2\), which means multiplying \( (x+h) \, \times \, (x+h) \).
  • Distribute each term: \( x \times x = x^2 \), \( x \times h = xh \), \( h \times x = hx \), \( h \times h = h^2 \).
Adding these results gives the expanded form \(x^2 + 2xh + h^2 \). After expansion, simplify further by distributing any factors outside the parentheses into the newly expanded terms.
Algebraic Simplification
Algebraic simplification involves combining like terms and making an expression as simple as possible. After expanding, we have the expression: \[ f(x+h) = 2(x^2 + 2xh + h^2) - 5(x+h) + 1 \].

Let's distribute the constants through the expression to simplify it step by step:
  • First, multiply \( 2 \) by each term inside \( (x^2 + 2xh + h^2) \): \( 2x^2 + 4xh + 2h^2 \).
  • Next, distribute \( -5 \) through \( (x+h) \): \(-5x - 5h \).
Combine these results with the constant \( 1 \) to obtain:
\[ f(x+h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \].

No like terms remain, so this is the simplified form of \( f(x+h) \). Ensuring each part of the expression is accurately expanded and simplified is critical for correct results. This process highlights the importance of careful arithmetic and consistent checking.

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

A 5 -foot-long board is leaning against a wall so that it meets the wall at a point 4 feet above the floor. What is the slope of the board? [Hint: Draw a picture.]

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$

ATHLETICS: Cardiovascular Zone Your maximum heart rate (in beats per minute) may be estimated as 220 minus your age. For maximum cardiovascular effect, many trainers recommend raising your heart rate to between \(50 \%\) and \(70 \%\) of this maximum rate (called the cardio zone). a. Write a linear function to represent this upper limit as a function of \(x\), your age. Then write a similar linear function to represent the lower limit. Use decimals instead of percents. b. Use your functions to find the upper and lower cardio limits for a 20 -year-old person. Find the cardio limits for a 60 -year-old person.

BUSINESS: Straight-Line Depreciation Straight-line depreciation is a method for estimating the value of an asset (such as a piece of machinery) as it loses value ( \({ }^{\prime \prime}\) depreciates" \()\) through use. Given the original price of an asset, its useful lifetime, and its scrap value (its value at the end of its useful lifetime), the value of the asset after \(t\) years is given by the formula: $$ \begin{aligned} \text { Value }=(\text { Price })-&\left(\frac{(\text { Price })-(\text { Scrap value })}{(\text { Useful lifetime })}\right) \cdot t \\ & \text { for } 0 \leq t \leq(\text { Useful lifetime }) \end{aligned} $$ a. A newspaper buys a printing press for $$\$ 800,000$$ and estimates its useful life to be 20 years, after which its scrap value will be $$\$ 60,000$$. Use the formula above Exercise 63 to find a formula for the value \(V\) of the press after \(t\) years, for \(0 \leq t \leq 20\) b. Use your formula to find the value of the press after 10 years. c. Graph the function found in part (a) on a graphing calculator on the window \([0,20]\) by \([0,800,000] .\) [Hint: Use \(x\) instead of \(t\).]

ECONOMICS: Per Capita Personal Income In the short run, per capita personal income (PCPI) in the United States grows approximately linearly. In 2001 PCPI was \(30.4\), and in 2009 it had grown to \(39.2\) (both in thousands of dollars). a. Use the two given (year, PCPI) data points \((1,30.4)\) and \((9,39.2)\) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2000 and \(y=\mathrm{PCPI}\). b. Use your linear relationship to predict PCPI in 2020 .
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.