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Dr. Antonio R. Quesada Director
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Lesson Plan
Introduction to Graphical and Algebraic Representations of Inverses
Lesson Summary: The student will complete an inquiry based lesson to discover the need for finding
inverses of functions. Emphasis is placed on several real life applications. Special attention is given to
using proper function notation and definitions.
Key Words:
Inverse
Inverse Functions
Graph Inverses
Algebraic Inverses
Background Knowledge:
Know and understand the definition of a function
Use proper function notation
Understand function vocabulary including inputs, outputs, and domain
A basic understanding of swapping inputs and outputs to generate inverses
Recognize reflections over various lines including y = 0, x = 0, y = x
Ohio Academic Content Standards:
Strand-Patterns, Functions and Algebra Standards
Benchmark-Grades 8-10 (C and D)
Learning Objectives:
Sketch inverses by hand
Visualize an inverse given a function
Use proper function and inverse notation
Generate an algebraic representation of the inverse of a simple function
Gain an appreciation for an inverse via a practical application
Materials:
A graphic calculator
Suggested Procedures: Prior to handing out the inquiry based lesson, the teacher must present the
basics of finding an inverse given a function. This must include swapping the inputs and outputs of
several functions (i.e.
3 f 7
7  f 1  3 ).
Attention Getter:
Use the DRAW Mode on the graphic calculator to draw inverses.
Group Size:
3 or less
Assessments:
Students must occasionally check answers with the teacher
Results will be checked at the end of the activity
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Dr. Antonio R. Quesada Director
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An Introduction to Graphical and Algebraic Representations of Inverses
Team Number 12:
Michelle Augusta
Connie Kramer
Michael Scott
Kim Snyder
Investigation Goals:
Sketch inverses by hand
Visualize an inverse given a function
Use proper function and inverse notation
Generate an algebraic representation of the inverse of a simple function
Gain an appreciation for an inverse via a practical application
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Dr. Antonio R. Quesada Director
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Solutions to
An Introduction to Graphical and Algebraic Representations of
Inverses
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Dr. Antonio R. Quesada Director
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An Introduction to Graphical and Algebraic Representations of Inverses
1
As we learned yesterday, if f is a function, then the effect of its inverse, denoted f , is to swap the role of its
inputs and outputs. Yesterday we learned the effect of the inverse upon specific input – output combinations.
Today we will learn two ways to represent the inverse of a function that will allow us to quickly gain information
about any desired input – output combination. First, though, do these exercises to review yesterday’s work.
Review Exercises
Suppose that both f and
f
1
are functions, and that the following are true:
f (3)  7 ;
f (2)  3 ;
f 1 (7)  2
f (7)  5 ;
If possible, answer the questions that follow. If it is not possible to know the answer to a question using the given
information, write “not enough information.”
1)
f
1
(5)
f 1 (7)
2)
-7
4)
3
f ( 2)
7
1
(3)
not enough information
f (7 )
5)
f
3)
f
6)
not enough information
1
(3)
-2
Graphs of Inverses
7)
Complete the following table for the function
x
f(x)
8)
0
-7
1
0
f ( x)  x 3  6 x 2  12 x  7
2
1
3
2
4
9
With the help of your graphing calculator, sketch the graph of f below. Be sure that the five points listed
in the table are included on your graph.
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9)
(a)
Dr. Antonio R. Quesada Director
Using the table you made in #7 and the skills you learned yesterday, answer the following questions.
f 1 (7)
f 1 (0)
(b)
0
10)
(c)
1
f
Make a table for
1
x
f (x)
f 1 (1)
(d)
2
f 1 (2)
(e)
3
f 1 (9)
4
whose content reflects the questions and answers from #9
-7
0
-1
11)
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0
1
1
2
2
3
9
4
Make an observation about the relationship between the tables for f and
this observation again later.
f
1
. You will be asked to use
The rows corresponding to input and output have been swapped
12)
Use the five points in the table of
f
1
from #10 as a guide to make a sketch of
f
1
, connecting the points
in a way that makes sense, on the same graph you used to sketch f in #8.
13)
1
Using the processes from #7-12, sketch the graph of g ( x)  2 and its inverse, g . Be sure to complete
both tables below and sketch two graphs, and that each graph contains all points from its respective table.
x
x
-1
0
1
2
3
g(x)
½
1
2
4
8
x
½
1
2
4
8
g-1(x)
-1
0
1
2
3
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14)
Dr. Antonio R. Quesada Director
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Looking at the two pairs of graphs you have made of a function with its inverse, make an observation about
the geometric relationship between a function and its inverse. [Hint: draw the line y  x along with each
pair of graphs]
The graph of a function and its inverse are symmetric over the line y=x
[Depending on their backgrounds, students’ observations may not be so sophisticated]
15)
Your calculator has a built – in feature that sketches the inverse of a function. Let’s use this feature to
verify our graphs from #8. To get started, type the rule for
f ( x)  x  6 x  12 x  7 into Y1. Next,
nd
return to the home screen. Press the DRAW button (2 – PRGM). Now you are on the DRAW menu.
Either press 8 or scroll down and press ENTER on #8. Either way, you should now see “DrawInv” on your
home screen. Do not press ENTER yet. Now you must tell the calculator of what function to draw the
inverse. Remember, our function rule is entered into Y 1, so make Y1 appear on the home screen
(remember, we do this via the VARS button). Now you should see “DrawInv Y1” on your home screen.
Press ENTER. Does it verify your graphs?
Yes
3
16)
Repeat the process from #15 to verify the graphs of g and
2
g 1 from #13.
I have successfully verified them.
17)
Using your calculator exclusively (that is, do not use a table), make a rough sketch of
h( x)  x 2  4 x  6
1
and h (using DrawInv) in the same xy-plane. Remember, for an undistorted representation, choose a
square window by pressing ZOOM:5 (If you change the window after performing DrawInv, you will have to
redo the DrawInv)
18)
Reread your observation from #14. Is the relationship between the above curves consistent with your
observation? If not, adjust your observation below.
Answers will vary
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19)
Dr. Antonio R. Quesada Director
1
Judging from your sketch of the graph of h , is h
1
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a function? Write a sentence to justify your answer.
No. It fails the vertical line test.
(Alternatively, there exists at least one value of x (say, x=3) with two outputs (namely, 1 and 3))
20)
h (that is, without generating a graph for h 1 ), make a conjecture as to how
1
you could have anticipated that h would not be a function.
Looking only at the graph of
The existence of a horizontal line along which two points lie for the function h becomes the existence of a
vertical line along which two points lie for the relation h-1
Sometimes, the inverse of a function is not, itself, a function. This can lead to some mathematical problems
for which there are strategies to handle. We will explore these problems and strategies in a later lesson.
A Real – World Example
21)
If you are driving a car and need to stop in the shortest distance possible, you will “slam” your brakes.
Evidence from your tires will be left on the road of the distance it has taken you to stop, in the form of a skid
mark. If two people are driving similarly – sized cars and slam their breaks, will each of their cars create
skid marks that are the same length? What do you think is the major factor upon which the length of their
respective skid marks depends?
No; the length of a set of skid marks will depend chiefly on the speed of the car that generates it.
22)
The length of a skid mark that is created by a car depends most on the speed of the car when the break is
applied. (Is this what you answered above?) Under normal conditions, the following function m roughly
models the length of a skid mark, in feet, generated by a car whose speed is x , in miles per hour:
m( x)  0.057 x 2
Using your calculator, generate a graph of this function. Think again about what x represents in this real –
world situation. Given this, think about whether any portion of the graph gives information that does not
make sense. Give the new, restricted domain of this function by eliminating all values of x that do not
make sense (and only those that do not make sense) given what x represents. Justify your response.
[If you need a hint, think about what you would be finding if you computed m(5) ]
The values of x must be nonnegative because speed must be nonnegative. So the domain is [0, ∞)
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23)
Roughly sketch the graph of m , with its restricted domain, below. (You may need to adjust your window
on your calculator…use the units below as a guide)
24)
Compute m( 40) and write a complete sentence to interpret its meaning (include units with your numbers)
91.2 feet. The skid mark created by a car traveling at a speed of 40 mph whose brakes are slammed will
be approximately 91.2 feet.
25)
An officer arriving at the scene of an accident measures the skid marks generated by a car to be 65 feet.
Why might the function m from above help the officer estimate the speed of the driver when brakes were
slammed? Why is it difficult to use the function m directly to find this speed?
The function relates speed to length of skid mark, but it takes speed as its input and produces length of
skid mark as its output. We do not have a speed to input, so we must work backwards.
26)
The function m has its input, x , measure speed in miles per hour and its output, m(x ) , measure length of
skid mark in feet. It would be much more convenient for us to have a function that swaps these roles.
a)
If we made a new function that swapped these roles, what would the input to this new function
measure? What would its output measure?
Input:
length of skid mark, in feet
Output: speed of car when brakes are slammed, in miles per hour
b)
What is the name for the mathematical concept that will have this effect on a function? What will
we symbolically call our new function in this particular case?
Inverse; m-1
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27)
Dr. Antonio R. Quesada Director
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As you learned to earlier, use the DrawInv feature on your calculator to draw the inverse of m . Given the
restriction that we had to impose on m in #22, explain why the entire graph generated by DrawInv is not
the inverse of m .
DrawInv drew the inverse of the portion of the function that was restricted, as well.
28)
Which portion of the graph generated by DrawInv should be discarded? Justify your answer using what
was learned earlier about how the graph of an inverse is generated.
The portion of the graph below the x-axis (with negative y-values), as this corresponds to the portion of the
domain that was restricted (with negative x-values).
1
29)
Sketch the graph of m below, taking into account your answer from #27. Be sure to label your axes with
the correct units. (You may have to adjust your viewing window)
30)
Using your sketch from #29, roughly estimate the speed of the car that generated the skid mark of 65 feet.
Explain the procedure you used to arrive at this estimate.
31)
Approximately 34 feet; by drawing the vertical line x = 65 and drawing a perpendicular from its point of
intersection with the graph of m to the y-axis.
Do you trust the accuracy of your estimate from #30 to, say, be within one-hundredth (.01) of a mile per
hour? What makes it difficult to obtain such accuracy using a graph?
No; the above graphical method has too much inexactness associated with it, from our sketch of the graph
to the thickness of the curves drawn.
Algebraic Representations of Inverses
32)
Let’s try to get a more exact answer to the question from #30. Remember, for the function m , the input,
x , represents a speed in miles per hour and the output, m(x) represents a skid mark length, in feet. If we
know that the length of a skid mark produced by a car is 65 feet, which variable do we know the value of,
x or m(x) ? What is the value of this variable?
m(x); m(x) = 65
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33)
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m is the equation from above that relates the variables x and m(x) .
2
Remember that m( x)  0.057 x . This equation has two variables, x and m(x ) . Using your answer
The algebraic representation of
from #32, remove the variable you know the value of, replace it with its value, and write your new equation
below.
65 = 0.057x2
34)
We have learned strategies for solving the type of equation that you wrote down in #33. Showing your
algebraic work, solve this equation. Use your calculator to round your final answer to the nearest
hundredth.
x≈  33.77
35)
Algebraically, your equation from #33-34 should have had two solutions. (Did you get two solutions?)
Below, state the two solutions, tell which solution should be discarded and explain why.
33.77 or -33.77.
-33.77 should be discarded, as it is not feasible as a speed (speed cannot be negative)
(Alternatively, we discarded this point as a point on the graph of m -1 earlier)
36)
37)
Repeat the processes from above to find the speed, to the nearest hundredth of a mile per hour, of cars
that make skid marks of 36 feet and 117 feet. Show your setups and algebraic work.
36 = 0.057x2
117 = 0.057x2
x ≈ 25.13 miles per hour
x ≈ 45.31 miles per hour
x . Go back to the original algebraic
representation for m , m( x)  0.057 x . Instead of replacing m(x ) with a specific value, as you did in
#34 and #36, let’s leave the equation as it is, but still solve for x . [Hint: The only difference will be that,
m (x )
after the first step, you won’t be able to compute the left side, it will have to remain
].
.057
Each time you find a speed, notice that you algebraically solve for
2
x=
38)
m(x)
0.057
Just as in #34 and #36, the equation above has two solutions. Explain why it was correct to discard one of
the solutions.
Because x represents speed, and speed cannot be negative (we discarded negative values of speed
earlier)
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39)
You should now have an equation that is solved for x , the speed in miles per hour, as a function of m(x ) ,
the length of the skid mark, in feet. Check your equation with me before continuing.
40)
The equation from #39 is a function, whose input is length of skid mark in feet, and whose output is speed
in miles per hour. It does not look like a function the way we normally see functions, because the variables
are not assigned as normal. For instance, normally, the input is assigned the variable name x . To begin
to correct this problem, rewrite your equation from #39, but remove the variable names x and m(x ) .
Replace them with sets of parentheses.
(
41)
)=
(___)
0.057
Take your equation from #39 and insert x into the parentheses that were previously occupied by m(x ) .
This renames the input using the normal input variable, x . Into the parentheses that were previously
occupied by
x , insert m 1 ( x) . State what the variables x and m 1 ( x) represent (include their units).
x
0.057
Equation:
m-1(x) =
x:
length of skid marks, in feet
m 1 ( x) :
speed of car when brakes are slammed, in miles per hour
1
We could have given any name to our new function, but we used m to remind ourselves that this function
is the inverse of m . Notice that in #41 we swapped the roles of input and output, the same procedure we
learned yesterday and used earlier when we were graphing inverses.
42)
On your calculator, generate a graph of your function from #41. How does it differ from the graph you
obtained by performing the DrawInv feature on the original function m ?
It only contains the portion of the graph on and above the x-axis
Extend: Try to explain this difference. [Hint: See your answer from #38]
DrawInv includes both solutions to the equation from #38, the inverse of the function with unrestricted
domain. The function from #41 is the inverse of m with its restricted domain.
43)
1
Now that we have an algebraic representation of m , we can use this to quickly and accurately find the
length of the skid mark generated by a car traveling at a given speed. Use your algebraic representation of
m 1 from #41 to find the following to the nearest hundredth of a mile per hour:
(a)
m 1 (65)
33.77 mph
(b)
m 1 (36)
25.13 mph
(c)
m 1 (117)
45.31 mph
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(d)
44)
Dr. Antonio R. Quesada Director
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Do your answers for (a) – (c) agree with their respective counterparts from #35 – 36?
Yes
1
1
To show that you understand what the functions m and m represent, compute m(50) and m (50) to
the nearest hundredth, and write a complete sentence to interpret the meaning of each (include the units of
the numbers in your sentences).
m(50)  142.5 feet
Sentence: A car traveling at a speed of 50 mph at the moment its brakes are slammed will produce a skid
mark of approximately 142.5 feet.
m 1 (50)  29.62 miles per hour
Sentence: A car that produced a skid mark of 50 feet was traveling approximately 29.62 miles per hour at
the moment its brakes were slammed.
45)
Find an algebraic representation for the inverse of the following functions using the procedure from #37 #41. Show each step of your work. (Note that, unlike the example done in #37 - #41, this process does
not always result in two solutions; neither of the examples below have two solutions).
(a)
f ( x)  3x  12
f-1(x) =
x  12
3
(b)
g-1(x) =
g ( x)  x 3  5
3
x5
46)
Using the skills you learned earlier in the lesson, try to think of a method you could use to verify that your
answers from #45 are correct.
47)
One way to convince yourself that your answers from #45 are correct is to use the DrawInv feature on your
calculator (is this what you thought of?). Type the algebraic rule for f from #45a into Y1. Draw its inverse
1
using DrawInv. Now, type the algebraic rule you found for f into Y2. If your answer is correct, it should
generate the same graph that was generated by DrawInv. Are you confident that your answer from #45a is
correct? Repeat this process for #45b. Are you confident that it is correct? (If not, try to find your
algebraic errors and repeat this process until you are confident in their correctness)
Answers will vary, but, hopefully, most will be “Yes”