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Mathematics LibreTexts

1.4 Composition of Functions

Suppose we wanted 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 the average daily temperature depends on the particular day of the year.  Notice how we have just defined two relationships: The temperature depends on the day, and the cost depends on the temperature.  Using descriptive variables, we can notate these two functions.

The first function, C(T), gives the cost C of heating a house when the average daily temperature is T degrees Celsius, and the second, T(d), gives the average daily temperature of a particular city on day d of the year.  If we wanted to determine the cost of heating the house on the 5th day of the year, we could do this by linking our two functions together, an idea called composition of functions.  Using the function T(d), we could evaluate T(5) to determine the average daily temperature on the 5th day of the year.  We could then use that temperature as the input to the C(T) function to find the cost to heat the house on the 5th day of the year:  C(T(5)).

 

Definition: Composition of Function

When the output of one function is used as the input of another, we call the entire operation a composition of functions.  We write f(g(x)), and read this as “f of g of x” or “f composed with g at x”. 

An alternate notation for composition uses the composition operator: File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image002.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image004.gif is read “f of g of x” or “f composed with g at x”,  just like  f(g(x)).

Example 1

Suppose c(s) gives the number of calories burned doing s sit-ups, and s(t) gives the number of sit-ups a person can do in t minutes.  Interpret c(s(3)).

When we are asked to interpret, we are being asked to explain the meaning of the expression in words.  The inside expression in the composition is s(3).  Since the input to the s function is time, the 3 is representing 3 minutes, and s(3) is the number of sit-ups that can be done in 3 minutes.  Taking this output and using it as the input to the c(s) function will gives us the calories that can be burned by the number of sit-ups that can be done in 3 minutes.

Note that it is not important that the same variable be used for the output of the inside function and the input to the outside function.  However, it is essential that the units on the output of the inside function match the units on the input to the outside function, if the units are specified.

Example 2

Suppose f(x) gives miles that can be driven in x hours, and g(y) gives the gallons of gas used in driving y miles.  Which of these expressions is meaningful: f(g(y)) or g(f(x))?

The expression g(y) takes miles as the input and outputs a number of gallons.  The function f(x) is expecting a number of hours as the input; trying to give it a number of gallons as input does not make sense.  Remember the units have to match, and number of gallons does not match number of hours, so the expression f(g(y)) is meaningless.

The expression f(x) takes hours as input and outputs a number of miles driven.  The function g(y) is expecting a number of miles as the input, so giving the output of the f(x) function (miles driven) as an input value for g(y), where gallons of gas depends on miles driven, does make sense.  The expression g(f(x)) makes sense, and will give the number of gallons of gas used, g, driving a certain number of miles, f(x), in x hours.

Try it Now

1. In a department store you see a sign that says 50% off of clearance merchandise, so final cost C depends on the clearance price, p, according to the function C(p). Clearance price, p, depends on the original discount, d, given to the clearance item, p(d).

Interpret C(p(d)).

Composition of Functions using Tables and Graphs

When working with functions given as tables and graphs, we can look up values for the functions using a provided table or graph, as discussed in section 1.1. We start evaluation from the provided input, and first evaluate the inside function. We can then use the output of the inside function as the input to the outside function. To remember this, always work from the inside out.

 

Example 3

Using the tables below, evaluateFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image006.gifand File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image008.gif

Text Box: x	f(x)
1	6
2	8
3	3
4	1


Text Box: x	g(x)
1	3
2	5
3	2
4	7

To evaluateFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image006.gif, we start from the inside with the value 3. We then evaluate the inside expressionFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image012.gifusing the table that defines the function g: File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image014.gif.  We can then use that result as the input to the f function, soFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image012.gif is replaced by the equivalent value 2 and we getFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image016.gif.  Then using the table that defines the function f, we find thatFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image018.gif.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image020.gif.

 

To evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image008.gif, we first evaluate the inside expression File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image022.gifusing the first table: File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image024.gifThen using the table for g we can evaluate:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image026.gif

Try it Now

2. Using the tables from the example above, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image028.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image030.gif.

 

Example 4

Using the graphs below, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image028.gif.

                             g(x)                                                             f(x)

Description: Description: (x-3)^2-1        Description: Description: -(x-3)^2+6

To evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image028.gif, we again start with the inside evaluation.  We evaluateFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image036.gif using the graph of the g(x) function, finding the input of 1 on the horizontal axis and finding the output value of the graph at that input.  Here, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image038.gif.  Using this value as the input to the f function, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image040.gif.  We can then evaluate this by looking to the graph of the f(x) function, finding the input of 3 on the horizontal axis, and reading the output value of the graph at this input.  Here, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image042.gif, soFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image044.gif.

 

Try it Now

3. Using the graphs from the previous example, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image046.gif.

Composition using Formulas

When evaluating a composition of functions where we have either created or been given formulas, the concept of working from the inside out remains the same.  First we evaluate the inside function using the input value provided, then use the resulting output as the input to the outside function.

 

Example 5

Given File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image048.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image050.gif, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image052.gif.

 

Since the inside evaluation isFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image054.gifwe start by evaluating the h(x) function at 1:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image056.gif

 

Then File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image058.gif, so we evaluate the f(t) function at an input of 5:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image060.gif

Try it Now

4. Using the functions from the example above, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image062.gif.

While we can compose the functions as above for each individual input value, sometimes it would be really helpful to find a single formula which will calculate the result of a composition f(g(x)).  To do this, we will extend our idea of function evaluation.  Recall that when we evaluate a function likeFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image048.gif, we put whatever value is inside the parentheses after the function name into the formula wherever we see the input variable. 

 

Example 6

GivenFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image048.gif, evaluateFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image064.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image066.gif.

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image068.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image070.gif

 

We could simplify the results above if we wanted to

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image072.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image074.gif

We are not limited, however, to using a numerical value as the input to the function. We can put anything into the function: a value, a different variable, or even an algebraic expression, provided we use the input expression everywhere we see the input variable. 

 

Example 7

Using the function from the previous example, evaluate f(a)

This means that the input value for t is some unknown quantity a.  As before, we evaluate by replacing the input variable t with the input quantity, in this case a.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image076.gif

The same idea can then be applied to expressions more complicated than a single letter.

 

Example 8

Using the same f(t) function from above, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image078.gif.

Everywhere in the formula for f where there was a t, we would replace it with the input File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image080.gif.  Since in the original formula the input t was squared in the first term, the entire input File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image082.gif needs to be squared when we substitute, so we need to use grouping parenthesesTo avoid problems, it is advisable to always use parentheses around inputs.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image084.gif

We could simplify this expression further to File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image086.gif if we wanted to:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image088.gif                 Use the “FOIL” technique (first, outside, inside, last)

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image090.gif             distribute the negative sign  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image092.gif              combine like terms

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image094.gif

Example 9

Using the same function, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image096.gif.

Note that in this example, the same variable is used in the input expression and as the input variable of the function. This doesn’t matter – we still replace the original input t in the formula with the new input expression, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image098.gif.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image100.gif

Try it Now

5. Given File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image102.gif, evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image104.gif.

 

This now allows us to find an expression for a composition of functions. If we want to find a formula for f(g(x)), we can start by writing out the formula for g(x).   We can then evaluate the function f(x) at that expression, as in the examples above.

 

Example 10

Let File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image106.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image108.gif, find f(g(x)) and g(f(x)).

Solution

To find f(g(x)), we start by evaluating the inside, writing out the formula for g(x)

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image108.gif

We then use the expression File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image110.gif as input for the function f.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image112.gif

We then evaluate the function f(x) using the formula for g(x) as the input.

Since File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image106.gif then File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image114.gif

This gives us the formula for the composition:  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image116.gif

Likewise, to find g(f(x)), we evaluate the inside, writing out the formula for f(x)

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image118.gif

Now we evaluate the function g(x) using x2 as the input.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image120.gif

 

Try it Now

6. Let File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image122.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image124.gif, find f(g(x)) and g(f(x)).

 

Example 11

A city manager determines that the tax revenue, R, in millions of dollars collected on a population of p thousand people is given by the formula File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image126.gif, and that the city’s population, in thousands, is predicted to follow the formula File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image128.gif, where t is measured in years after 2010.  Find a formula for the tax revenue as a function of the year.

Solution

Since we want tax revenue as a function of the year, we want year to be our initial input, and revenue to be our final output.  To find revenue, we will first have to predict the city population, and then use that result as the input to the tax function.  So we need to find R(p(t)).  Evaluating this,

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image130.gif

This composition gives us a single formula which can be used to predict the tax revenue during a given year, without needing to find the intermediary population value. 

For example, to predict the tax revenue in 2017, when t = 7 (because t is measured in years after 2010)

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image132.gifmillion dollars

In some cases, it is desirable to decompose a function – to write it as a composition of two simpler functions.

Example 12

Write File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image134.gif as the composition of two functions.

 

We are looking for two functions, g and h, so File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image136.gif.  To do this, we look for a function inside a function in the formula for f(x).  As one possibility, we might notice that File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image138.gif is the inside of the square root.  We could then decompose the function as:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image140.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image142.gif

We can check our answer by recomposing the functions:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image144.gif

Note that this is not the only solution to the problem.  Another non-trivial decomposition would be File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image146.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image148.gif

Important Topics of this Section

  • Definition of Composition of Functions
  • Compositions using:
    • Words
    • Tables
    • Graphs
    • Equations

 

Try it Now Answers

1.      The final cost, C, depends on the clearance price, p, which is based on the original discount, d.  (Or the original discount d, determines the clearance price and the final cost is half of the clearance price.)

2.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image150.gif          and      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image152.gif

3.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image154.gif

4.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image156.gif      did you remember to insert your input values using parentheses?

5.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image158.gif  

6.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image160.gif

     File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image162.gif

Section 1.4 Exercises

 

Given each pair of functions, calculate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image164.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image166.gif.

1. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image168.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image170.gif                     2. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image172.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image174.gif

3. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image176.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image178.gif                  4. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image180.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image182.gif

 

Text Box: x		 
 

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

Use the table of values to evaluate each expression

5.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image187.gif    

6.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image189.gif

7.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image191.gif  

8.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image193.gif

9.      File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image195.gif   

10.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image197.gif

11.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image199.gif

12.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image201.gif

 

Description: Description: GraphsUse the graphs to evaluate the expressions below.

13.  Description: Description: GraphsFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image006.gif

14.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image208.gif

15.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image210.gif   

16.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image166.gif

17.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image213.gif   

18.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image215.gif  

19.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image199.gif

20.  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image218.gif

 


 

For each pair of functions, find File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image220.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image222.gif.  Simplify your answers.

21. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image224.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image226.gif                    22. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image228.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image230.gif

23. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image232.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image234.gif                  24. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image236.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image238.gif

25. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image240.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image242.gif             26. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image244.gif , File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image246.gif

27.  If  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image248.gif,File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image250.gifand File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image252.gif, find  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image254.gif

 

28.  If  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image256.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image258.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image260.gif , find  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image254.gif 

 

29.  Given functions File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image263.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image265.gif, state the domains of the following functions using interval notation.

a.       Domain of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image267.gif  

b.      Domain of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image269.gif

c.       Domain of  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image271.gif

 

30.  Given functions File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image273.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image275.gif, state the domains of the following functions using interval notation.

a.       Domain of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image277.gif  

b.      Domain of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image279.gif

c.       Domain of  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image281.gif

 

31.  The function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image283.gif gives the number of items that will be demanded when the price is p. The production cost, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image285.gif is the cost of producing x items. To determine the cost of production when the price is $6, you would do which of the following:

a. Evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image287.gif                     b. Evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image289.gif

c. Solve File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image291.gif                     d. Solve File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image293.gif

 

32.  The function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image295.gif gives the pain level on a scale of 0-10 experienced by a patient with d milligrams of a pain reduction drug in their system.  The milligrams of drug in the patient’s system after t minutes is modeled by File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image297.gif.  To determine when the patient will be at a pain level of 4, you would need to:

a. Evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image299.gif                   b. Evaluate File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image301.gif            

c. Solve File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image303.gif                    d. Solve File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image305.gif

 

33.  The radius r, in inches, of a spherical balloon is related to the volume, V, by File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image307.gif.  Air is pumped into the balloon, so the volume after t seconds is given by File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image309.gif.

a.       Find the composite function  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image311.gif

b.      Find the time when the radius reaches 10 inches.

 

34.  The number of bacteria in a refrigerated food product is given by File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image313.gif, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image315.gif,  where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image317.gif, where t is the time in hours.

a.       Find the composite function  File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image319.gif

b.      Find the time when the bacteria count reaches 6752

 

Find functions File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image321.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image323.gif so the given function can be expressed as File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image325.gif.

35. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image327.gif                                         36. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image329.gif

37. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image331.gif                                             38. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image333.gif

39. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image335.gif                                     40. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image337.gif

41.  Let File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image321.gif be a linear function, with form File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image340.gif for constants a and b.  [UW]

a.       Show that File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image342.gif is a linear function

b.      Find a function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image323.gif such that File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image345.gif

 

42.  Let File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image347.gif  [UW]

a.       Sketch the graphs of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image349.gif on the interval −2 ≤ x ≤ 10.

b.      Your graphs should all intersect at the point (6, 6). The value x = 6 is called a fixed point of the function f(x)since File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image351.gif; that is, 6 is fixed - it doesn’t move when f is applied to it. Give an explanation for why 6 is a fixed point for any function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image353.gif.

c.       Linear functions (with the exception of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image355.gif) can have at most one fixed point. Quadratic functions can have at most two. Find the fixed points of the function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image357.gif.

d.      Give a quadratic function whose fixed points are x = −2 and x = 3.

 

43.  A car leaves Seattle heading east. The speed of the car in mph after m minutes is given by the function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image359.gif.   [UW]

a.       Find a function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image361.gif that converts seconds s into minutes m. Write out the formula for the new function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image363.gif; what does this function calculate?

b.      Find a function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image365.gif) that converts hours h into minutes m. Write out the formula for the new function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image367.gif; what does this function calculate?

c.       Find a function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image369.gif that converts mph s into ft/sec z. Write out the formula for the new function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image371.gif; what does this function calculate?

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)