Inverse Functions Module
In this set of activities we introduce inverses of functions. This is done in every precalculus text, and
generally done well. The approach in this course is quite different, and can be done by itself or with a
standard exposition of the topic. The activities look at inverse functions as elements of a group. By
defining a group on a subset of the set of standard precalculus functions (SPF’s) we can get all of the
standard information about inverse functions.
The Inverse module can be taught with the Groups Module or after the Groups module.
Rationale for this Presentation:
1) This presentation allows us to view functions as elements of a set, as well as machines that
input and output numbers.
2) The author’s personal feeling that recognizing a group helps to “organize” your thoughts about a
set.
3) It connects the different elements of the course: We saw groups of symmetry functions on
logos and now we see a group of standard precalculus functions.
4) Students see an additional example of group.
Activity 1 Inverse functions
Goals:
1) Students think of STP’s as elements of a set.
2) Using the operation of composition students work to identify a group structure
3) Students can identify an identity element.
Introductions to inverses on standard precalculus functions
(This introduction is written for the instructor and student)
In the groups module we saw that a set of elements formed a group if it had
1) an associative operation
2) the set was closed under the operation
3) an identity element
4) an inverse element for each element
In the following activities we’ll examine the set of SPF’s and see if we can find a group structure in it or a
subset of it. We use the operation of composition. Note that we are viewing these SPF’s as elements of
a set.
Notation
Taking the functions to be elements of a set we’ll sometimes write
g ◦f= g and , instead of f ◦g(x)=g(x)
Finding an identity element for this group
Notice that if f(x)=x then for many functions g ◦f =g and , f ◦g =g
Exercise1
Part 1
Let f(x)=x. Student should pick their own SPF that is defined on all real numbers. ( i.e avoid
squareroots, etc). Have students perform the two compositions
a) g ◦f (x)
b) f ◦g (x)
c) g ◦f (1)
d) f ◦g (1)
Part 2
A sample of 4-6 students should put their work on the board from part 1.
1) Based on the results of part 1, what can be said about composing functions defined on the real
line with the function f(x)=x.
(The class should conclude that g ◦f (x)=g(x) and f ◦g (x)=g(x))
Suggestions for Exercise 1 Part 1
1) Picking functions. Picking an arbitrary function and applying it to an equation, theorem, or
result is a technique that the course tries to consistently develop. The goal is to get students to
realize the general abstract nonsense that is usually seen in a mathematics course. If students
have difficulty choosing a function, direct them to the standard precalculus functions that are
defined on the entire real line(linear, quadratic, exponential, etc.)
2) Machine models. Although students might have skills to do composition without the help of the
machine model it’s still important to bring it back every now and then. The machine model
allows students to understand the operation of composition. This understanding will help them
in the future. For example in a calculus class it will help them understand the chain rule.
Suggestions for Exercise 1 Part 2
1) Comparing results. The boardwork is key to this exercise. Students should be able to compare
their results to other students’ results. The statements they make in this exercise should reflect
the work on the board.
2) Precise statements. Ask students to develop their statements until they are precise statements.
It’s important that the class develops statements and not the instructor. The instructor’s job is
to act as a moderator for the class.
Exercise 2
Recall that an element of a group e is an identity element if for each a belonging to the group
a#e=a and e#a=a
1) Using the results of exercise 1 and the definition of an identity element state why f(x)=x is a
good candidate for an identity element . Note that it’s a good candidate when we restrict our
functions to set of SPF defined on the set of real numbers.
Suggestions for Exercise 2
1) Identity elements. Students should keep score, so far we have a set,
a) A set. The set of all functions defined on the real line.
b) An operation. The operation is composition.
c) An identity element. f(x)=x.
2) Group. Students should keep in mind that we’re looking to find a group structure on some set of
functions. The instructor should emphasize that we are looking at functions in a different way.
We are trying to see them as elements of groups.
Exercise 3 Problems with Domains
f(x)=x is a good candidate for an identity element on the set of functions defined on all real numbers
let’s see what happens when we compose it with a function that is not defined set of real numbers.
1) Let f(x)=x and g(x)=√𝑥. Draw the machine model for g ◦f (-1) and , f ◦g (-1)
i.e. the machine model is something that looks like what’s below. For the exercise use -1 instead of x.
xxg(x)
xg(x)g(x)
1
2) Let f(x)=x and g(x)=𝑥 . What’s g ◦ f(0). Use the machine model to find it ?
Suggestions for Exercise 3
1) Domains. Exercise 3 shows that to make a group out of a set of functions you must consider the
domains of the functions. Our candidate for an identity element doesn’t make sense in the
cases used above. Domain is an issue.
Remarks for the class and instructor:
From Exercise 2 we see that f(x)=x is a pretty good candidate for being the identity element in a
group of SPF’s. Exercise 3 shows that we run into some domain issues when we start using
certain functions.
From now on as we’re trying to find a group in the set of SPF’s we’ll restrict ourselves to those
SPF’s whose domain is all real numbers. Using this set of functions we can definitely say that g ◦f
(x)= g(x) and , f ◦g (x)=g(x) where f(x)=x .
Summary
(for instructors and students)
Recall that we started looking at the set of all SPF’s trying to find a group in this collection. We had to
restrict the set to a subset , the set of all SPF’s that whose domain is the set of all real numbers because
we wanted to make a meaningful identity element. Below is a list of things we’ve found and things we
need.
What do we have now
1) A subset of our set original set , all SPF’s defined on all real numbers
2) An operation, composition.
3) Composition is associative so we have that.
4) An identity element , f(x)=x
What do we need
1) We need to check that the group is closed.
2) We need to make sure that each element has an inverse.
Activity 2
Defining Inverses
Goals:
1) Students can transfer the definition of an inverse element in a group to the standard
precalculus concept.
2) Students can show that g ◦g-1 (x)=x and , g-1 ◦g (x)=x for the set of STP’s defined on all real
numbers functions.
Introductions:
(This is for students and instructors)
Review of the previous lecture
In the previous activity we were trying to find a subset of the SPF’s that could form a group. We
found the function f(x)=x was a good candidate for an identity element but found that there were
domain issues that came up with some functions. In order to use f(x)=x as the identity element we
decided to restrict ourselves to functions that were defined on the set of real numbers. We found that
the function f(x)=x made a good identity element for this set, meaning for any other function in our set
g(x),
g ◦f (x)= g(x) and , f ◦g (x)=g(x)
Important Notation:
a) We’ll call f our identity element ID. So instead of f(x)=x we’ll write ID(x)=x.
b) Also taking the functions to be elements of a set we’ll sometimes write
g ◦f= g and , instead of f ◦g(x)=g(x)
Looking for Inverses
Now that we have an identity we can go about identifying inverses of elements. This leads to a further
restriction of the set of functions. All the functions in the set must have inverses in order to be a group.
So we’ll have to restrict our set to SPF’s defined on the all real numbers that have inverses. Additionally
the inverses have to be defined on the entire real line.
There’s a problem, we don’t know what it means to be an inverse in this context. The group theory
definition is as follows
Let a and b belong to the set. b is an inverse of a if a#b=e and b#a=e where e is the identity
More Notation:
a)
the inverse of a is notated as “a-1”. We can rephrase the definition above the following way
a#a-1=e and a-1#a=e
Key Identity
For our set of functions the identity element is called ID. Relating the definition of an inverse elements
of a group to functions would mean a function and it’s inverse would have to satisfy the following
equations,
g-1◦g(x)=ID(x)=x and g ◦ g-1(x)=ID(x)=x
or
g ◦ g-1=ID and g-1 ◦ g=ID
Definition: We’ll say that the function h is an inverse of the function g if
h◦g(x)=ID(x)=x and g ◦ h(x)=ID(x)=x
or
g ◦ h=ID and h◦ g=ID
Exercise 1
𝑥−3
2
Let g(x)=2x+3 and h(x)=
Show that g ◦ h=ID and h ◦ g=ID. Conclude that g and h are inverses of each other by
referring to the definition.
2) Show that g ◦ h(3)=3 and h ◦ g(3)=3
1)
Suggestions for Exercise 1
1) Testing functions. Through calculations the students can test whether a function is the inverse
of another. Students should be aware that they are testing these two functions.
2) Abstract nonsense. The definitions will seem like abstract nonsense to students. Part 2 of the
exercise will help students understand what the symbols f◦ g=id means. It will be helpful to do
more computations like the one in part 2 so students can put the abstract nonsense in context.
Exercise 2
Part 1
𝑥−3
.
2
Let g(x)=2x+3 and h(x)=
1)
2)
3)
4)
5)
6)
In the previous exercises we found that these functions were inverses.
What’s g(3) , and h(9)
What’s g(1) , and h(5)
What’s g(-1) , and h(1)
What’s h(7) , and g(2)
What’s h(-7) , and g(-5)
What’s h(11) , and g(4)
Part 2 (Class Discussion)
1) What kind of statement can you make about the relationship between the inputs and outputs of
h(x) and g(x). Base your statement on the results that you found in part 1 of the exercise.
Suggestions for Exercise 2
1)
Statements. Students should make a statement looks similar to the following one.
If h is the inverse of g, then g(x)=y implies h(y)=x .
Also if h(p)=q this implies g(q)=p
We can think of the machine as follows
XYX
YXY
2) Precise Statements. Students will not make precise statements at first. Write their imprecise
statements on the board and ask the class to refine them. Students should do the work
composing the statement. The instructor should act as the moderator for the class.
Exercise 3
Use the statements from exercise 2 to answer the following questions. Take two functions g and h
where h is the inverse of g.
1) If g(5)=6 , what’s h(6)?
2) If g(1)=3 , what’s h(3)?
3) If h(-3)=20 , what’s g(20)?
4) If h(-7)=14 , what’s g(14)?
Suggestions for Exercise 3
1) Abstract nonsense. Exercise 3 tries to transfer the abstract statements made in exercise 2 into a
real situation. A goal of the course is for students to naturally create examples that illustrate
general abstract nonsense statements.
2) Conceptual understanding. If the students don’t understand the exercise then they don’t
understand a fundamental concept of inverses. More work deriving examples and applications
of the statement from exercise 2 is needed.
Summary
Keep in mind that we’re trying to find a subset of the set of SPf’s that forms a group . The following tells
us how far we’ve come on the road to finding these inverses.
What do we have:
The group of functions that we have has the following properties
1) We’re working with functions with inverses defined on all real numbers. Our operation is
composition.
2) The identity of the group is ID which is defined by ID(x)=x
3) Each function must have an inverse to be in our group. The function h is inverse of the function
g if g ◦ h=ID and h◦ g=ID
4) Since composition is associative we’re fine there.
What we need
1) Is the group closed. Is the composition of two functions defined on all real numbers a function
that’s defined on all real numbers.
Activity 3
Which Functions have Inverses?
Goals :
1) Students can use the horizontal line test to see if a function has an inverse.
2) Students realize how the horizontal line test is related to properties of functions and their
inverses.
Introduction
(This is for students and instructors)
In the previous activity we found a definition for inverses. Using the definition of inverses of a
function we’ll be able to find a quick test to tell whether a function has an inverse or not. We’ll also find
a quick way to compute the equation of the inverse of a function.
When does a function have an inverse?
Recall Part 2 of exercise 3 from the previous activity. We found that the
if g is the inverse of f then f(x)=y implies g(y)=x. also if g(p)=q that
implies that f(q)=p (property 1)
Example 1
Take the functions g(x)=2x+3 and h(x)=
𝑥−3
2
. In the last activity we showed that these functions were
inverses. Then we did some exercise to show if g(3)=9 the h(9)=3 and more generally if g(x)=y then
h(y)=x
Notice that not all functions have this property. The following example illustrates this.
Example 2
Let f(x)=x2 . f(2)=4 this implies that f-1(4)=2. Sound’s o.k. but f(-2)=4 so that implies f-1(4)= -2. f-1 (4) = 2
and -2. So f-1 is not a function since it has two outputs for one input. This implies f(x)=x2 doesn’t have
an inverse.
Exercise 1
(For this exercise students should simply try to repeat the concept illustrated by the example above)
1) Use property 1 to show that f(x)=x4 cannot have an inverse.
2) Use property 1 to show that f(x)=x2-2 cannot have an inverse.
Remark:
From the example and the exercise above we have the following:
It must be the case that if an inverse exists then any two different inputs of the original function must
have different outputs.
What does this statement mean graphically?
Exercise 2
Part 1
1)
2)
3)
4)
Have the students split up into groups. Groups should work to graph the function f(x)=x2 .
Find the points (2,f(2)) and (-2,f(-2)) on your graph.
Find the points (2,f(2)) and (-2,f(-2)) on your graph.
Find the points (2,f(2)) and (-2,f(-2)) on your graph.
Part 2
(The instructor should put students’ thoughts on the board, and prompt them to make more and more
exact statements about the issue.)
1) Characterize what happens graphically when two inputs of a function have the same output.
(to the instructor: the students should say something along the lines of the horizontal line test.)
Part 3
1) Sketch the graph of a function in which no points of the graph lie on one horizontal line.
2) Sketch the graph of a function in which two points of the graph lie on one horizontal line.
(Have 4-5 students put up their sketches.Leave the sketches on the board for the next part of the
exercise.)
Part 4
1) Suppose the graph of a function doesn’t have any two points lying on the same horizontal lines.
Can two inputs of the function have the same output?
2) Make a statement that summarizes the results of part1-4.
Review of statement so far.
1) Let f have an inverse call it f-1 then, if f(x)=y implies f-1(y)=x
2) If a function has two inputs that give the same output then the function cannot have an inverse.
3) If a function has two inputs that have the same output it doesn’t pass the horizontal line test.
To Do Finish the construction of the horizontal line test
Suppusoe you pass this horizontal line test this implies a inverse functions exist. a method for creating
the inverse functionis as follows if f(x)=y then the inverse g will have g(y)=x. This works only if we pass
the horizontal line test.
Now sketch the graph of a function that passes this horizontal line test
A function has an inverse if and only if it passes the Horizontal Line Test.
Definition: Horizontal Line test a function has an inverse if when you put a horizontal line through the
graph of it the line hits at most 1 point of the graph
Exercise 2
Which of the graphs below pass the horizontal line test.
1) Insert 4 graphs a) Examine the graphs of functions below. Which of the functions have inverses.
Insert graphs of x3 , x4 , x 5 , x6
b) Based on the results from part a do you think f(x)=x65 has an inverse.
c) Make a statement about functions with inverses based on the information given above.
2) Draw a function that passes the horizontal line test. Draw one that doesn’t.
Activity 4
Finding the equation for the inverse of a function
Goals:
1) Students learn an algorithm for finding the inverse of a function.
2) Students learn the concept of finding the inverse of a function on a restricted domain.
3) Students can describe the relationship between the graphs of a function and it’s inverse.
If a function has an inverse can I find what if is?
If the function passes the horizontal line test it has an inverse. We can find it’s inverse by using the
following algorithm:
1) Solve the equation for x
2) Where you see an x change it to f(x)-1 and where you see an f(x) change it to x.
Exercise 1
1) What’s the inverse of f(x)=3x + 5, call it f-1
2) Show that f ◦ f-1=x and f-1 ◦f(x)=x
Exercise 2
Part 1
1) Split the class up into groups. Each group should find the inverse of one of the following
functions.
a) f(x)=3x—6
5
b) f(x)=√𝑥 − 3
c) f(x)=x5+ 3
d) f(x)=3x3- 5
Part 2
2) Have the class put the graphs of the functions on the board. Ask them to characterize the
relationship between the graphs of a function and it’s inverse.
Remark
The graph of the inverse of a function is sketched by reflecting the graph of the original function over
the line y=x
Exercise 3
Below are the graphs of three functions
Insert (square root function, and the exponential function in different forms)
Sketch the inverses of these functions on the same set of axis. Where are points corresponding to the
labeled points located?
Homework
1) Find the inverses of the following functions:
a) g(x)=-3x + 7
3
b) h(x)= √3𝑥 + 7
5
c) j(x)= √𝑥 + 3
1) For each function and it’s inverse that you found show that g ◦ g-1(x)=x and g-1 ◦ g(x)=x.
2) Write down the equation for a linear function f(x) . Then find it’s inverse. Graph the function
and it’s inverse.
3) Here are the graphs of three functions with that have inverse. Draw the inverses of each
function on it’s respective graph. Where are points corresponding to the labeled points
located?
4) Draw the graph of a function that passes the horizontal line test . Sketch the graph of the
inverse of that function.