Course: 6.8 具有函数的操作

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具有函数的操作操作
Your office is having a holiday party. The number of people at the party can be modeled by $\begin{aligned} g (x) = - 20 x^{2} + 80 x + 20 \end{aligned}$ where x is the time in hours. T he number of appetizers being offered can be modeled by $\begin{aligned} f (x) = - 100 x^{4} + 800 x^{3} - 2400 x^{2} + 3200 x - 600 \end{aligned}$ . We can find how many appetizers are available for each person at a certain point in time during the party by dividing the functions. In this section, we discuss operations on functions— addition , subtraction , multiplication , and division .
::您的办公室正在举行假日聚会。党内的人数可以用 g( x)\\\\ 20x2+80x+20为模范, 其中x是小时中的时数。提供的开胃菜的数量可以用 f( x)\\\ 100x4+800x3-2400x3+3200x2+3200x- 600为模范。我们可以通过划分功能来找到在党内某个时间点为每个人提供多少开胃菜。在此节中, 我们讨论功能的附加、减、乘法和分割操作。



Operations on Functions
::函数上的操作

Addition and Subtraction
::加和减

The rules for adding and subtracting functions are below.
::增加和减去功能的规则如下。

Addition and Subtraction
::加和减

Addition: $\begin{aligned} (f + g) (x) = f (x) + g (x) \end{aligned}$
::加f+g)(x)=f(x)+g(x)

Subtraction: $\begin{aligned} (f - g) (x) = f (x) - g (x) \end{aligned}$
::减法f)-g(x)=f(x)-g(x)

Example 1
::例1

If $\begin{aligned} f (x) = | x - 4 | + 5 \end{aligned}$ and $\begin{aligned} g (x) = \sqrt{x + 2} - 7 \end{aligned}$ , what is $\begin{aligned} f + g \end{aligned}$ ?
::如果 f( x) x- 45 和 g( x) = x+2 - 7, 什么是 f+g ?

Solution: We can add the functions by adding the rule for each.
::解决方案:我们可以通过为每种功能增加规则来增加功能。

$\begin{aligned} (f + g) (x) & = f (x) + g (x) \\ = | x - 4 | + 5 + \sqrt{x + 2} - 7 \\ = | x - 4 | + \sqrt{x + 2} - 2 \end{aligned}$

:f+g)(x) = f(x) +g(x) +g(x) = x(x) = x(x) =- 4) = 5+x+2 -7* x - 4* =x+2-2

The only like terms are the numbers 5 and -7, so we can add them to get -2.
::唯一相似的术语是5号和7号, 所以我们可以加上它们来得到 -2号。

by Mathispower4u demonstrates how to add and subtract functions.
::以 Mathispower4u 演示如何增减函数。

Example 2
::例2

If $\begin{aligned} f (x) = x + 5 \end{aligned}$ and $\begin{aligned} g (x) = x^{2} - 4 x + 8 \end{aligned}$ , what is $\begin{aligned} f - g \end{aligned}$ ?
::如果f(x)=x+5和g(x)=x2-4x+8,什么是f-g?

Solution: First, we replace f and g with their rules. Then, s ince these are both , we follow the rules for subtracting polynomials.
::解决方案:首先,我们用他们的规则取代f和g。然后,由于这两条规则都是,我们遵循了减去多面体的规则。

$\begin{aligned} (f - g) (x) & = f (x) - g (x) \\ = (x + 5) - (x^{2} - 4 x + 8) \\ = x + 5 - x^{2} + 4 x - 8 \\ = - x^{2} + 5 x - 3 \end{aligned}$

:g)(x)=f(x)-g(x)=(x+5)-(x2-4x+8)=x+5-x2+4x-8*x2+4x-8*x2+5x3)

WARNING
::警告

Recall that $\begin{aligned} f (x) \end{aligned}$ is not multiplication by x .
::回顾 f( x) 不是 x 乘法。

For example, in Example 2 we would not subtract f and g and then multiply by x .
::例如,在例2中,我们不会减去f和g,然后乘以x。

Multiplication and Division
::计算和司

The rules for multiplication and division are below.
::乘法和除法规则如下。

Multiplication and Division
::计算和司

Multiplication: $\begin{aligned} (f g) (x) = f (x) \cdot g (x) \end{aligned}$
::乘法fg)(x)=f(x)=g(x)

Division: $\begin{aligned} (\frac{f}{g}) (x) = \frac{f (x)}{g (x)}, g (x) \neq 0 \end{aligned}$
::项数 : (fg) (x) = f(x) g(x) x, g(x) =0

Example 3
::例3

If $\begin{aligned} f (x) = 5 x^{- 1} \end{aligned}$ and $\begin{aligned} g (x) = 4 x + 7 \end{aligned}$ , what is $\begin{aligned} f g \end{aligned}$ ?
::如果f(x)=5x-1和g(x)=4x+7,什么是fg?

Solution: We have two options here. We can leave the -1 exponent as is or rewrite it as a positive exponent in the denominator. We present both options.
::解答: 我们有两个选项。我们可以保留 -1 的标注, 或重写它作为分母中积极的标注。我们提出两个选项。

$\begin{aligned} (f g) (x) & = f (x) \cdot g (x) \\ = 5 x^{- 1} (4 x + 7) \\ = 20 x^{0} + 35 x^{- 1} distribute and use the product rule of exponents \\ = 20 + 35 x^{- 1} x^{0} = 1 \\ OR \\ (f g) (x) & = f (x) \cdot g (x) \\ = \frac{5}{x} (4 x + 7) \\ = \frac{20 x + 35}{x} \end{aligned}$

:g)(x) =f(x) =f(x) =5x1(4x+7) =20x0+35x-1分布并使用 expronents=20+35x-1x0=1OR(fg)(x) =f(x) =5x(4x+7) =20x+35xx

Both representations are correct.
::这两种陈述都是正确的。

by CK-12 demonstrates how to perform function operations.
::CK-12 显示如何执行功能操作。

Example 4
::例4

If the number of people at a party can be modeled by $\begin{aligned} g (x) = - 20 x^{2} + 80 x + 20 \end{aligned}$ and t he number of appetizers being offered can be modeled by $\begin{aligned} f (x) = - 100 x^{4} + 800 x^{3} - 2400 x^{2} + 3200 x - 600 \end{aligned}$ , where x is the time in hours, w hat function models how many appetizers are available for each person at a certain point in time during the party?
::如果可以以 g(x) 20x2+80x+20 来模拟党内的人数,而且提供开胃菜的数量可以用 f(x) 100x4+800x3-2400x2+3200x-600来模拟,那么x是小时的小时数,那么在党内某个时间点,每个人有多少开胃菜可以使用什么功能模型?

Solution: The words "for each" imply that we need to divide the functions to find the function to model this situation.
::解决方案:“对每个人”一语意味着我们需要将功能分开,以找到能模拟这种情况的功能。

$\begin{aligned} (\frac{f}{g}) (x) & = \frac{f (x)}{g (x)} \\ = \frac{- 100 x^{4} + 800 x^{3} - 2400 x^{2} + 3200 x - 600}{- 20 x^{2} + 80 x + 20} \\ = \frac{- 20 (5 x^{4} - 40 x^{3} + 120 x^{2} - 160 x + 30)}{- 20 (x^{2} - 4 x - 1)} factor out greatest common factor - 20 \\ = \frac{5 x^{4} - 40 x^{3} + 120 x^{2} - 160 x + 30}{x^{2} - 4 x - 1} \end{aligned}$

:fg)(x) = f(x) g(x) = f(x) f(x) = @100x4+8004+8003-2400x2+3200x-600-202x2+80x+20=20=2020(5x4-40x3+120x2-160x+30)-20(x2-4x-1) =f(x) f(x) =100x4+100x4+8003-2400x2+3200x2-300x600-20x2+80x+20x20=20x3+120x2-160x+30x) -20(x2-4x-1) =最大共同系数-2020=5x4x4-40x3+120x2x2-160x3+120x2x-4x-1)

We can also express this after doing polynomial long division . Then, $\begin{aligned} (\frac{f}{g}) (x) = 5 x^{2} - 20 x + 45 + \frac{75}{x^{2} - 4 x - 1} \end{aligned}$ .
::我们也可以在做多面长的分割后表达这一点。然后, (fg(x) = 5x2-20x+45+75x2-4x-1) 。

Exponents
::指数

Raising a function to a power is analogous to raising a number to power.
::将职能提升为权力类似于将若干职能提升为权力。

Exponents
::指数

There are two different notations for raising exponents to powers.
::提出权力的推手有两种不同的称谓。

$\begin{aligned} f^{n} (x) & [f (x)]^{n} \\ except when x = - 1 & all powers \end{aligned}$
For example, $\begin{aligned} f^{2} (x) = [f (x)]^{2} = f (x) \cdot f (x) \end{aligned}$
::fn( x) [f( x)]n 除非当 x\\\\\ x]n 时, x}1all powers 例如, f2( x) = [f( x)]2= f( x) f( x) f( x) f( x)

* $\begin{aligned} f^{- 1} (x) \end{aligned}$ is how we indicate inverse functions, which will be discussed in a later chapter.
::* f-1(x)是我们如何表示反向功能,将在后一章讨论。

Example 5
::例5

Find $\begin{aligned} f^{4} (x) \end{aligned}$ when $\begin{aligned} f (x) = \sqrt{x - 1} \end{aligned}$ .
::f( x) =x- 1 时查找 f4( x) 。

Solution: We apply the exponent to f and take advantage of the rules of working with square roots.
::解决办法:我们运用推手,利用与平方根合作的规则。

$\begin{aligned} f^{4} (x) & = f (x) \cdot f (x) \cdot f (x) \cdot f (x) \\ = \sqrt{x - 1} \cdot \sqrt{x - 1} \cdot \sqrt{x - 1} \cdot \sqrt{x - 1} \\ = \sqrt{(x - 1) (x - 1) (x - 1) (x - 1)} \\ = \sqrt{(x - 1)^{4}} \\ = (x - 1)^{2} \\ = x^{2} - 2 x + 1 \end{aligned}$

::f4(x) = f(x)xx) f(x) f(x) f(x) f(x) f(x) f(x) =x- 1(x)x- 1(x)x- 1(x)x- 1(x)x- 1(x)1(x)-(x)-1(x)-1(x) =(x- 1) 4(x) 4=(x-1) 2=x2-2x+1

Feature: Turning a Profit
::特征: 创造利润

by Deirdre Mundy
::由Deirdre Mundy 编辑

You have just started a new business. You are excited as you deposit that first payment from a client into your bank account. The profits are rolling in, right? Don't celebrate too quickly. You might have a long way to go before your company is actually producing a profit.
::你刚开始一个新业务。当你把客户的首笔付款存入你的银行账户时, 您很兴奋。利润正在滚动, 对不对? 不要太快庆祝。您可能还有很长的路要走, 您的公司才能真正产生利润。

Costs and Benefits
::费用和福利

Profit can be described by the function $\begin{aligned} P = R - C \end{aligned}$ , where $\begin{aligned} P \end{aligned}$ stands for profit, $\begin{aligned} R \end{aligned}$ represents revenue or earnings, and $\begin{aligned} C \end{aligned}$ represents total costs. The functions that represent revenue and costs vary from product to product and company to company. However, costs always include some combination of employee salaries and benefits, the costs of raw materials, overhead such as rent and utilities, and taxes owed to the local, state, and federal government.
::P=R-C是利润,R代表收入或收入,C代表总成本,收入和成本的功能因产品和公司而异,但成本总是包括雇员工资和福利、原材料成本、租金和公用事业等间接费用以及欠地方、州和联邦政府的税款。

Your company earns a profit when the money it takes in exceeds all of these expenses. If you are not taking in enough money to cover your expenses, you are losing money. If your earnings equal your expenses, you are at point where your business is "breaking even." It can take a long time for a company to break even. For instance, Amazon did not turn a profit until it had been in business for nearly a decade. A company that is only breaking even cannot protect itself against future downturns. It cannot increase the size of its workforce, give employees raises, or expand into new markets. If a company is not making a profit, it cannot grow.
::当你的公司所花的钱超过所有这些开支时,它就会赚取利润。如果你没有花足够的钱来支付开支,你就会失去钱。如果你的收入等于你的开支,你就会处于你的生意“扯平 ” 的地步。公司要倒闭需要很长的时间。例如,亚马逊公司在经营近十年之后才有利润。一个仅仅崩溃的公司甚至无法保护自己免受未来衰退的影响。它不能增加劳动力规模,增加员工,或向新市场扩张。如果一个公司没有盈利,它就不能增长。

When a company does make a profit, it can use the money for one of several things. It can pay down debts, it can reinvest the money in the business, or it can pay the money to its owners. If a company is publicly traded, its owners are the people who have bought stock in that company. When a stock pays dividends, it is paying out a portion of its profits to the people who own its stock.
::当一个公司盈利时,它可以将钱用于几件事情之一。它可以偿还债务,可以将钱再投资于企业,也可以将钱还给所有者。如果一个公司是公开交易,其所有者就是购买该公司股票的人。当一个股票支付股息时,它向拥有其股票的人支付其利润的一部分。

For an analysis of comparing the costs and the revenue of a business and how they relate to prate to profit, watch by lostmy1.
::为了分析比较企业的成本和收入以及它们与利率和利润的关系,请看《迷途》1。

Summary
::摘要

To add, subtract, multiply or divide functions, replace the functions with their rules and perform the operations.
::添加、减、乘或分割功能,用其规则替换功能,并进行操作。

Review
::回顾

For problems 1-5, use the following functions to perform the operations.
::对于问题1-5,使用下列功能来进行作业。

$\begin{aligned} f (x) = x^{2} + 5 g (x) = 3 \sqrt{x - 5} h (x) = 5 x + 1 \end{aligned}$

:x)=x2+5g(x)=3x-5h(x)=5x+1

1. $\begin{aligned} f + h \end{aligned}$
::1. f+h

2. $\begin{aligned} h - g \end{aligned}$
::2.h-g

3. $\begin{aligned} g^{2} \end{aligned}$
::3. 克2

4. $\begin{aligned} \frac{f}{g} \end{aligned}$
::4. fg

5. $\begin{aligned} f h \end{aligned}$
::5. 时

For problems 6-10, use the following functions to perform the operations.
::对于问题6-10,使用下列功能来进行作业。

$\begin{aligned} p (x) = \frac{5}{x} q (x) = 5 \sqrt{x} r (x) = \frac{\sqrt{x}}{5} s (x) = \frac{1}{5} x^{2} \end{aligned}$

::p(x)=5xq(x)=5xr(x)=x5s(x)=15x2

6. $\begin{aligned} p s \end{aligned}$
::6. Ps 数

7. $\begin{aligned} \frac{q}{r} \end{aligned}$
::7. qrr

8. $\begin{aligned} q + r \end{aligned}$
::8. q+r

9. $\begin{aligned} r - p \end{aligned}$
::9. r-p

10. $\begin{aligned} s^{3} \end{aligned}$
::10. 3 s3

Explore More
::探索更多

1. The area of a rectangle is $\begin{aligned} 2 x^{2} \end{aligned}$ . The length of the rectangle is $\begin{aligned} \sqrt{x + 3} \end{aligned}$ . What is the width of the rectangle? What restrictions, if any, are on this value?
::1. 矩形区域为 2x2. 矩形的长度为 x+3 。矩形的宽度是多少?对于这个值有什么限制(如果有的话)?

2. The revenue for a business can be modeled by $\begin{aligned} R (x) = 85 x \end{aligned}$ and the costs for the business can be modeled by $\begin{aligned} C (x) = 25 x + 2500 \end{aligned}$ , where x is the number of units sold. What is the profit function? What is the break-even point?
::2. 企业收入可以R(x)=85x为模型,企业成本可以C(x)=25x+2500为模型,售出单位数量为x。利润的作用是什么?平衡点是什么?

3. Is $\begin{aligned} f^{2} (x) \end{aligned}$   the same as $\begin{aligned} f (x^{2}) \end{aligned}$ ? Consider what happens when $\begin{aligned} f (x) = x + 4 \end{aligned}$ .
::3. f2(x) 是否与 f(x2) 相同? 考虑当 f(x) =x+4 时会发生什么。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

/

Section outline

General

具有函数的操作操作

Operations on Functions ::函数上的操作

Addition and Subtraction ::加和减

Addition and Subtraction ::加和减

Example 1 ::例1

Example 2 ::例2

WARNING ::警告

Multiplication and Division ::计算和司

Multiplication and Division ::计算和司

Example 3 ::例3

Example 4 ::例4

Exponents ::指数

Exponents ::指数

Example 5 ::例5

Feature: Turning a Profit ::特征: 创造利润

Costs and Benefits ::费用和福利

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Operations on Functions
::函数上的操作

Addition and Subtraction
::加和减

Addition and Subtraction
::加和减

Example 1
::例1

Example 2
::例2

WARNING
::警告

Multiplication and Division
::计算和司

Multiplication and Division
::计算和司

Example 3
::例3

Example 4
::例4

Exponents
::指数

Exponents
::指数

Example 5
::例5

Feature: Turning a Profit
::特征: 创造利润

Costs and Benefits
::费用和福利

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX