For doll collectors, the Shirley Temple doll is a known quantity. Most collectors know that if you have one or can get one, these dolls can catch a pretty penny depending on the year and the condition.
The lovable, adorable Shirley Temple was a symbol of hope during the dark, despair-filled days during the Great Depression. The child actress had a stream of No. 1 box office hits in the 1930s.
In October 1934, Ideal Toy Corporation applied for a patent, and the Shirley Temple doll was officially announced in an issue of the retail industry magazine âPlaythings.â Shirley Temple dolls come in many varieties, but the most popular and lasting version has been the composition doll, first made when Shirley Temple was a star.
Different Varieties
The Shirley Temple dolls made in composition were fabricated from a composite of glue mixed with sawdust. This method was commonly used to make doll heads and limbs from the 1920s through the 1950s. The dolls had open-mouthed smiles and a choice of precious red, blond, and brunette curls. The dolls ranged in size from 11-inches to 27-inches tall, and they sold very well when they were new.
Later versions of Shirley Temple dolls were made in vinyl versions by Ideal. The vinyl versions made in 1957 are the most desirable to collectors. Some of these dolls were made up to 3-feet tall. These were the 'playpal' variety.
The 1990s saw porcelain commemorative dolls made in Templeâs likeness being sold as collectibles. The likeness of Temple had also been made into a paper doll many times.
Though the dolls have been produced since the 1930s, there have been several gaps in production, including in the 1960s and '70s.
Value of the Dolls
The 1930s composition-made dolls tend to craze and crack when not stored properly, so only dolls that are in excellent to mint condition bring the highest values. Rare or unusual outfits complete with an original doll can also add a good sum to the overall value.
Only mint-condition dolls in their original boxes can sell for $1,000 to $2,000, as can rarer varieties like the baby Shirley dolls.
The 1957 vinyl Shirley Temple dolls sell for much more than the later early 1970s Shirley dolls. The 1957 dolls average about $100. The price difference is determined by the size, the outfit, the condition, and if the original box still exists. The re-released Shirley Temple vinyl dolls in 1982 generally sell for less than $25 even if mint in the box.
Danbury Mint porcelain dolls from the 1990s resell at a wide range of prices from $10 to $150.
Manufacturers
The best-known company that produced the Shirley Temple dolls was Ideal. Ideal produced the dolls from the 1930s until the company went out of business in the early 1970s. Danbury Mint created most of the collectible porcelain versions of the doll.
Shirley Temple Collectors Club
In 1960, the first Shirley Temple Collectors Club was established. From the '60s through the '80s, new Shirley Temple dolls were regularly distributed by companies like Montgomery Ward and Danbury Mint, which was aimed primarily at nostalgic adult collectors.
Definition:To establish a selling price for a product
No matter what type of product you sell, the price you charge your customers or clients will have a direct effect on the success of your business. Though pricing strategies can be complex, the basic rules of pricing are straightforward:
Before setting a price for your product, you have to know the costs of running your business. If the price for your product or service doesn't cover costs, your cash flow will be cumulatively negative, you'll exhaust your financial resources, and your business will ultimately fail.
To determine how much it costs to run your business, include property and/or equipment leases, loan repayments, inventory, utilities, financing costs, and salaries/wages/commissions. Don't forget to add the costs of markdowns, shortages, damaged merchandise, employee discounts, cost of goods sold, and desired profits to your list of operating expenses.
Most important is to add profit in your calculation of costs. Treat profit as a fixed cost, like a loan payment or payroll, since none of us is in business to break even.
Because pricing decisions require time and market research, the strategy of many business owners is to set prices once and 'hope for the best.' However, such a policy risks profits that are elusive or not as high as they could be.
When is the right time to review your prices? Do so if:
Prices are generally established in one of four ways:
Cost-Plus Pricing
Many manufacturers use cost-plus pricing. The key to being successful with this method is making sure that the 'plus' figure not only covers all overhead but generates the percentage of profit you require as well. If your overhead figure is not accurate, you risk profits that are too low. The following sample calculation should help you grasp the concept of cost-plus pricing:
Demand Price
Demand pricing is determined by the optimum combination of volume and profit. Products usually sold through different sources at different prices--retailers, discount chains, wholesalers, or direct mail marketers--are examples of goods whose price is determined by demand. A wholesaler might buy greater quantities than a retailer, which results in purchasing at a lower unit price. The wholesaler profits from a greater volume of sales of a product priced lower than that of the retailer. The retailer typically pays more per unit because he or she are unable to purchase, stock, and sell as great a quantity of product as a wholesaler does. This is why retailers charge higher prices to customers. Demand pricing is difficult to master because you must correctly calculate beforehand what price will generate the optimum relation of profit to volume.
Competitive Pricing
Competitive pricing is generally used when there's an established market price for a particular product or service. If all your competitors are charging $100 for a replacement windshield, for example, that's what you should charge. Competitive pricing is used most often within markets with commodity products, those that are difficult to differentiate from another. If there's a major market player, commonly referred to as the market leader, that company will often set the price that other, smaller companies within that same market will be compelled to follow.
To use competitive pricing effectively, know the prices each competitor has established. Then figure out your optimum price and decide, based on direct comparison, whether you can defend the prices you've set. Should you wish to charge more than your competitors, be able to make a case for a higher price, such as providing a superior customer service or warranty policy. Before making a final commitment to your prices, make sure you know the level of price awareness within the market.
If you use competitive pricing to set the fees for a service business, be aware that unlike a situation in which several companies are selling essentially the same products, services vary widely from one firm to another. As a result, you can charge a higher fee for a superior service and still be considered competitive within your market.
Markup Pricing
Used by manufacturers, wholesalers, and retailers, a markup is calculated by adding a set amount to the cost of a product, which results in the price charged to the customer. For example, if the cost of the product is $100 and your selling price is $140, the markup would be $40. To find the percentage of markup on cost, divide the dollar amount of markup by the dollar amount of product cost:
$40 ? $100 = 40%
This pricing method often generates confusion--not to mention lost profits--among many first-time small-business owners because markup (expressed as a percentage of cost) is often confused with gross margin (expressed as a percentage of selling price). The next section discusses the difference in markup and margin in greater depth.
Pricing Basics
To price products, you need to get familiar with pricing structures, especially the difference between margin and markup. As mentioned, every product must be priced to cover its production or wholesale cost, freight charges, a proportionate share of overhead (fixed and variable operating expenses), and a reasonable profit. Factors such as high overhead (particularly when renting in prime mall or shopping center locations), unpredictable insurance rates, shrinkage (shoplifting, employee or other theft, shippers' mistakes), seasonality, shifts in wholesale or raw material, increases in product costs and freight expenses, and sales or discounts will all affect the final pricing.
Overhead Expenses. Overhead refers to all nonlabor expenses required to operate your business. These expenses are either fixed or variable:
Cost of Goods Sold. Cost of goods sold, also known as cost of sales, refers to your cost to purchase products for resale or to your cost to manufacture products. Freight and delivery charges are customarily included in this figure. Accountants segregate cost of goods on an operating statement because it provides a measure of gross-profit margin when compared with sales, an important yardstick for measuring the business' profitability. Expressed as a percentage of total sales, cost of goods varies from one type of business to another.
Normally, the cost of goods sold bears a close relationship to sales. It will fluctuate, however, if increases in the prices paid for merchandise cannot be offset by increases in sales prices, or if special bargain purchases increase profit margins. These situations seldom make a large percentage change in the relationship between cost of goods sold and sales, making cost of goods sold a semivariable expense.
Determining Margin. Margin, or gross margin, is the difference between total sales and the cost of those sales. For example: If total sales equals $1,000 and cost of sales equals $300, then the margin equals $700.
Gross-profit margin can be expressed in dollars or as a percentage. As a percentage, the gross-profit margin is always stated as a percentage of net sales. The equation: (Total sales ? Cost of sales)/Net sales = Gross-profit margin
Using the preceding example, the margin would be 70 percent.
($1,000 ? $300)/$1,000 = 70%
When all operating expenses (rent, salaries, utilities, insurance, advertising, and so on) and other expenses are deducted from the gross-profit margin, the remainder is net profit before taxes. If the gross-profit margin is not sufficiently large, there will be little or no net profit from sales.
Some businesses require a higher gross-profit margin than others to be profitable because the costs of operating different kinds of businesses vary greatly. If operating expenses for one type of business are comparatively low, then a lower gross-profit margin can still yield the owners an acceptable profit.
The following comparison illustrates this point. Keep in mind that operating expenses and net profit are shown as the two components of gross-profit margin, that is, their combined percentages (of net sales) equal the gross-profit margin:
Markup and (gross-profit) margin on a single product, or group of products, are often confused. The reason for this is that when expressed as a percentage, margin is always figured as a percentage of the selling price, while markup is traditionally figured as a percentage of the seller's cost. The equation is:
(Total sales ? Cost of sales)/Cost of sales = Markup
Using the numbers from the preceding example, if you purchase goods for $300 and price them for sale at $1,000, your markup is $700. As a percentage, this markup comes to 233 percent:
$1,000 ? $300 ? $300 = 233%
In other words, if your business requires a 70 percent margin to show a profit, your average markup will have to be 233 percent.
You can now see from the example that although markup and margin may be the same in dollars ($700), they represent two different concepts as percentages (233% versus 70%). More than a few new businesses have failed to make their expected profits because the owner assumed that if his markup is X percent, his or her margin will also be X percent. This is not the case.
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In this section, you will:
Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.
Using descriptive variables, we can notate these two functions. The functionC(T)
gives the costC
of heating a house for a given average daily temperature inT
degrees Celsius. The functionT(d)
gives the average daily temperature on dayd
of the year. For any given day,Cost=C(T(d))
means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperatureT(d).
For example, we could evaluateT(5)
What are the 3 stages of aml. to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would writeC(T(5)).
By combining these two relationships into one function, we have performed function composition, which is the focus of this section.
Combining Functions Using Algebraic Operations
Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
Suppose we need to add two columns of numbers that represent a husband and wifeâs separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that yearâs incomes and then collecting all the data in a new column. Ifw(y)
is the wifeâs income andh(y)
is the husbandâs income in yeary,
and we wantT
to represent the total income, then we can define a new function.
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
T=h+w
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functionsf(x)
andg(x)
with real number outputs, we define new functionsf+g,fâg,fg,
andfg
by the relations
(f+g)(x)=f(x)+g(x)(fâg)(x)=f(x)âg(x)(fg)(x)=f(x)g(x)(fg)(x)=f(x)g(x)
Find and simplify the functions(gâf)(x)
and(gf)(x),
givenf(x)=xâ1
andg(x)=x2â1.
Are they the same function?
Begin by writing the general form, and then substitute the given functions.
(gâf)(x)=g(x)âf(x)(gâf)(x)=x2â1â(xâ1)=x2âx=x(xâ1)(gf)(x)=g(x)f(x)(gf)(x)=x2â1xâ1=(x+1)(xâ1)xâ1 where xâ 1=x+1
No, the functions are not the same.
Note: For(gf)(x),
the conditionxâ 1
is necessary because whenx=1,
the denominator is equal to 0, which makes the function undefined.
Find and simplify the functions(fg)(x)
and(fâg)(x).
Are they the same function?
(fg)(x)=f(x)g(x)=(xâ1)(x2â1)=x3âx2âx+1(fâg)(x)=f(x)âg(x)=(xâ1)â(x2â1)=xâx2
No, the functions are not the same.
Create a Function by Composition of Functions
Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:
We read the left-hand side asâf
composed withg
atx,â
and the right-hand side asâf
ofg
ofx.â
The two sides of the equation have the same mathematical meaning and are equal. The open circle symbolâ
is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases f(g(x))â f(x)g(x).
It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the functiong
takes the inputx
first and yields an outputg(x).
Then the functionf
takesg(x)
as an input and yields an outputf(g(x)).
In general,fâg
andgâf
are different functions. In other words, in many casesf(g(x))â g(f(x))
for allx.
We will also see that sometimes two functions can be composed only in one specific order.
For example, iff(x)=x2
andg(x)=x+2,
then
f(g(x))=f(x+2)=(x+2)2=x2+4x+4
but
These expressions are not equal for all values ofx,
so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input valuex=â12.
Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.
When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any inputx
and functionsf
andg,
this action defines a composite function, which we write asfâg
such that
(fâg)(x)=f(g(x))
The domain of the composite functionfâg
is allx
such thatx
is in the domain ofg
andg(x)
is in the domain off.
It is important to realize that the product of functionsfg
is not the same as the function compositionf(g(x)),
because, in general,f(x)g(x)â f(g(x)).
Determining whether Composition of Functions is Commutative
Using the functions provided, findf(g(x))
andg(f(x)).
Determine whether the composition of the functions is commutative.
Letâs begin by substitutingg(x)
intof(x).
Now we can substitutef(x)
intog(x).
g(f(x))=3â(2x+1)=3â2xâ1=â2x+2
We find thatg(f(x))â f(g(x)),
so the operation of function composition is not commutative.
The functionc(s)
gives the number of calories burned completings
sit-ups, ands(t)
gives the number of sit-ups a person can complete int
minutes. Interpretc(s(3)).
The inside expression in the composition iss(3).
Because the input to the s-function is time,t=3
represents 3 minutes, ands(3)
is the number of sit-ups completed in 3 minutes.
Usings(3)
as the input to the functionc(s)
gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).
Supposef(x)
gives miles that can be driven inx
hours andg(y)
gives the gallons of gas used in drivingy
miles. Which of these expressions is meaningful:f(g(y))
org(f(x))?
The functiony=f(x)
is a function whose output is the number of miles driven corresponding to the number of hours driven.
The functiong(y)
is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:
number of gallons =g(number of miles)
The expressiong(y)
takes miles as the input and a number of gallons as the output. The functionf(x)
requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expressionf(g(y))
is meaningless.
The expressionf(x)
takes hours as input and a number of miles driven as the output. The functiong(y)
requires a number of miles as the input. Usingf(x)
(miles driven) as an input value forg(y),
where gallons of gas depends on miles driven, does make sense. The expressiong(f(x))
makes sense, and will yield the number of gallons of gas used,g,
driving a certain number of miles,f(x),
inx
hours.
Are there any situations wheref(g(y))andg(f(x))would both be meaningful or useful expressions?
Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order.
The gravitational force on a planet a distance r from the sun is given by the function G(r).
The acceleration of a planet subjected to any force F
is given by the function a(F).
Form a meaningful composition of these two functions, and explain what it means.
A gravitational force is still a force, so a(G(r))
makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F))
does not make sense.
Evaluating Composite Functions
Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner functionâs output as the input for the outer function.
Evaluating Composite Functions Using Tables
When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.
Using [link], evaluatef(g(3))
andg(f(3)).
To evaluatef(g(3)),
we start from the inside with the input value 3. We then evaluate the inside expressiong(3)
using the table that defines the functiong:
g(3)=2.
We can then use that result as the input to the functionf,
sog(3)
is replaced by 2 and we getf(2).
Then, using the table that defines the functionf,
we find thatf(2)=8.
To evaluateg(f(3)),
we first evaluate the inside expressionf(3)
using the first table:f(3)=3.
Then, using the table forg,â
we can evaluate
g(f(3))=g(3)=2
[link] shows the composite functionsfâg
andgâf
as tables.
andg(f(4))=g(1)=3
Evaluating Composite Functions Using Graphs
When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from thex-
and y-
axes of the graphs.
Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.
Using [link], evaluatef(g(1)).
To evaluatef(g(1)),
we start with the inside evaluation. See [link].
We evaluateg(1)
using the graph ofg(x),
finding the input of 1 on thex-
axis and finding the output value of the graph at that input. Here,g(1)=3.
We use this value as the input to the functionf.
We can then evaluate the composite function by looking to the graph off(x),
finding the input of 3 on the x-
axis and reading the output value of the graph at this input. Here,f(3)=6,
sof(g(1))=6.
[link] shows how we can mark the graphs with arrows to trace the path from the input value to the output value.
g(f(2))=g(5)=3
Evaluating Composite Functions Using Formulas
When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a compositionf(g(x)).
To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function likef(t)=t2ât,
we substitute the value inside the parentheses into the formula wherever we see the input variable.
Given a formula for a composite function, evaluate the function.
Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input
Givenf(t)=t2ât
andh(x)=3x+2,
evaluatef(h(1)).
Because the inside expression ish(1),
we start by evaluatingh(x)
at 1.
Thenf(h(1))=f(5),
so we evaluatef(t)
at an input of 5.
f(h(1))=f(5)f(h(1))=52â5f(h(1))=20
It makes no difference what the input variablest
andx
were called in this problem because we evaluated for specific numerical values.
Givenf(t)=t2ât
andh(x)=3x+2,
evaluate
Finding the Domain of a Composite Function
As we discussed previously, the domain of a composite function such asfâg
is dependent on the domain ofg
and the domain off.
It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such asfâg.
Let us assume we know the domains of the functionsf
andg
separately. If we write the composite function for an inputx
asf(g(x)),
we can see right away thatx Final fantasy tactics anime.
must be a member of the domain ofg
in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see thatg(x)
must be a member of the domain off,
otherwise the second function evaluation inf(g(x))
cannot be completed, and the expression is still undefined. Thus the domain offâg
consists of only those inputs in the domain ofg
that produce outputs fromg
belonging to the domain off.
Note that the domain off
composed withg
is the set of allx
such thatx
is in the domain ofg
andg(x)
is in the domain off.
The domain of a composite functionf(g(x))
is the set of those inputsx
in the domain ofg
for whichg(x)
is in the domain off.
**Given a function compositionf(g(x)),
determine its domain.**
Find the domain of
(fâg)(x) wheref(x)=5xâ1 and g(x)=43xâ2
The domain ofg(x)
consists of all real numbers exceptx=23,
since that input value would cause us to divide by 0. Likewise, the domain off
consists of all real numbers except 1. So we need to exclude from the domain ofg(x)
that value ofx
for whichg(x)=1.
So the domain offâg
is the set of all real numbers except23
and2.
This means that
xâ 23orxâ 2
We can write this in interval notation as
Finding the Domain of a Composite Function Involving Radicals
Find the domain of
Because we cannot take the square root of a negative number, the domain ofg
is(ââ,3].
Now we check the domain of the composite function
For (fâg)(x)=3âx+2,3âx+2â¥0,
since the radicand of a square root must be positive. Since square roots are positive, 3âxâ¥0,
or, 3âxâ¥0,
which gives a domain of (-â,3]
.
This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain offâg
can contain values that are not in the domain off,
though they must be in the domain ofg.
Find the domain of
[â4,0)âª(0,â)
Decomposing a Composite Function into its Component Functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.
Writef(x)=5âx2
as the composition of two functions.
We are looking for two functions,g
andh,
sof(x)=g(h(x)).
To do this, we look for a function inside a function in the formula forf(x).
As one possibility, we might notice that the expression5âx2
is the inside of the square root. We could then decompose the function as
We can check our answer by recomposing the functions.
g(h(x))=g(5âx2)=5âx2
Writef(x)=43â4+x2
as the composition of two functions.
Access these online resources for additional instruction and practice with composite functions.
Key Equation
Key Concepts
Section ExercisesVerbal
How does one find the domain of the quotient of two functions,fg?
Find the numbers that make the function in the denominatorg
equal to zero, and check for any other domain restrictions onf
andg,
such as an even-indexed root or zeros in the denominator.
If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.
Yes. Sample answer: Letf(x)=x+1 and g(x)=xâ1.
Thenf(g(x))=f(xâ1)=(xâ1)+1=x
andg(f(x))=g(x+1)=(x+1)â1=x.
Sofâg=gâf.
How do you find the domain for the composition of two functions,fâg?
Algebraic
Givenf(x)=x2+2x
andg(x)=6âx2,
findf+g,fâg,fg,
andfg.
Determine the domain for each function in interval notation.
(f+g)(x)=2x+6,
domain:(ââ,â)
(fâg)(x)=2x2+2xâ6,
domain:(ââ,â)
(fg)(x)=âx4â2x3+6x2+12x,
domain:(ââ,â)
(fg)(x)=x2+2x6âx2,
domain:(ââ,â6)âª(â6,6)âª(6,â)
Givenf(x)=â3x2+x
andg(x)=5,
findf+g,fâg,fg,
andfg.
Determine the domain for each function in interval notation.
Givenf(x)=2x2+4x
andg(x)=12x,
findf+g,fâg,fg,
andfg.
Determine the domain for each function in interval notation.
(f+g)(x)=4x3+8x2+12x,
domain:(ââ,0)âª(0,â)
(fâg)(x)=4x3+8x2â12x,
domain:(ââ,0)âª(0,â)
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(fg)(x)=x+2,
domain:(ââ,0)âª(0,â)
(fg)(x)=4x3+8x2,
domain:(ââ,0)âª(0,â)
Givenf(x)=1xâ4
andg(x)=16âx,
findf+g,fâg,fg,
andfg.
Determine the domain for each function in interval notation.
Given f(x)=3x2
andg(x)=xâ5,
findf+g,fâg,fg,
andfg.
Determine the domain for each function in interval notation.
(f+g)(x)=3x2+xâ5,
domain:[5,â)
(fâg)(x)=3x2âxâ5,
domain:[5,â)
(fg)(x)=3x2xâ5,
domain:[5,â)
(fg)(x)=3x2xâ5,
domain:(5,â)
Givenf(x)=x
andg(x)=|xâ3|,
findgf.
Determine the domain of the function in interval notation.
Givenf(x)=2x2+1
andg(x)=3xâ5,
find the following:
a. 3; b.f(g(x))=2(3xâ5)2+1;
c.f(g(x))=6x2â2;
d.(gâg)(x)=3(3xâ5)â5=9xâ20;
e.(fâf)(â2)=163
For the following exercises, use each pair of functions to findf(g(x))
andg(f(x)).
Simplify your answers.
f(g(x))=x2+3+2,g(f(x))=x+4x+7
f(g(x))=x+1x33=x+13x,g(f(x))=x3+1x
(fâg)(x)=12x+4â4=x2,(gâf)(x)=2xâ4
For the following exercises, use each set of functions to findf(g(h(x))).
Simplify your answers.
f(g(h(x)))=(1x+3)2+1
Givenf(x)=1x
andg(x)=xâ3,
find the following:
Givenf(x)=2â4x
andg(x)=â3x,
find the following:
Given the functionsf(x)=1âxxandg(x)=11+x2,
find the following:
Given functionsp(x)=1x
andm(x)=x2â4,
state the domain of each of the following functions using interval notation:
Given functionsq(x)=1x
andh(x)=x2â9,
state the domain of each of the following functions using interval notation.
Forf(x)=1x
andg(x)=xâ1,
write the domain of(fâg)(x)
in interval notation.
For the following exercises, find functionsf(x)
andg(x)
so the given function can be expressed ash(x)=f(g(x)).
sample:f(x)=x3g(x)=xâ5
sample:f(x)=4xg(x)=(x+2)2
sample:f(x)=x3g(x)=12xâ3
sample:f(x)=x4g(x)=3xâ2x+5
sample: f(x)=x
g(x)=2x+6
sample: f(x)=x3
g(x)=(xâ1)
sample: f(x)=x3
g(x)=1xâ2
sample: f(x)=x
g(x)=2xâ13x+4
Graphical
For the following exercises, use the graphs off,
shown in [link], andg,
shown in [link], to evaluate the expressions.
2
5
4
0
For the following exercises, use graphs off(x),
shown in [link],g(x),
shown in [link], andh(x),
shown in [link], to evaluate the expressions.
2
1
4
4
How To Price Original Composition Of GoldNumeric
For the following exercises, use the function values forf and g
shown in [link] to evaluate each expression.
|
x
</math></strong> |
f(x)
</math></strong> |
g(x)
</math></strong> || 0 | 7 | 9 || 1 | 6 | 5 || 2 | 5 | 6 || 3 | 8 | 2 || 4 | 4 | 1 || 5 | 0 | 8 || 6 | 2 | 7 || 7 | 1 | 3 || 8 | 9 | 4 || 9 | 3 | 0 |
9
4
2
3
For the following exercises, use the function values forf and g
shown in [link] to evaluate the expressions.
11
0
7
For the following exercises, use each pair of functions to findf(g(0))
andg(f(0)).
f(g(0))=27,g(f(0))=â94
f(g(0))=15,g(f(0))=5
For the following exercises, use the functionsf(x)=2x2+1
andg(x)=3x+5
to evaluate or find the composite function as indicated.
18x2+60x+51
gâg(x)=9x+20
Extensions
For the following exercises, usef(x)=x3+1
andg(x)=xâ13.
2
(ââ,â)
Letf(x)=1x.
For the following exercises, letF(x)=(x+1)5,
f(x)=x5,
andg(x)=x+1.
False
For the following exercises, find the composition whenf(x)=x2+2
for allxâ¥0
andg(x)=xâ2.
(fâg)(6)=6
; (gâf)(6)=6
(fâg)(11)=11,(gâf)(11)=11
Real-World Applications
The functionD(p)
gives the number of items that will be demanded when the price isp.
The production costC(x)
is the cost of producingx
items. To determine the cost of production when the price is $6, you would do which of the following?
The functionA(d)
gives the pain level on a scale of 0 to 10 experienced by a patient withd
milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patientâs system aftert
minutes is modeled bym(t).
Which of the following would you do in order to determine when the patient will be at a pain level of 4?
A store offers customers a 30% discount on the pricex
of selected items. Then, the store takes off an additional 15% at the cash register. Write a price functionP(x)
that computes the final price of the item in terms of the original pricex.
(Hint: Use function composition to find your answer.)
A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according tor(t)=25t+2,
find the area of the ripple as a function of time. Find the area of the ripple att=2.
A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formular(t)=2t+1,
express the area burned as a function of time,t
(minutes).
Use the function you found in the previous exercise to find the total area burned after 5 minutes.
The radiusr,
in inches, of a spherical balloon is related to the volume,V,
byr(V)=3V4Ï3.
Air is pumped into the balloon, so the volume aftert
seconds is given byV(t)=10+20t.
The number of bacteria in a refrigerated food product is given by N(T)=23T2â56T+1,
3<T<33,
whereT
is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t)=5t+1.5,
where t
is the time in hours.
Glossary
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