MAT 117 WEEK 5 LESSON PLAN
Total estimated time : 125 minutes
Objectives:
Composition of Functions
1. The algebra of functions
 Find the sum, difference, product, and quotient of functions.
 Form the composition of two functions and determine the domain.
 Identify a function as the composite of two functions.
 Use combinations and composition of functions to solve an application problem.
2. Inverse functions
 Determine if a function has an inverse.
 Find the inverse of a function.
 Graph a function and its inverse.
Motivation: [2 minutes]
In Calculus, it is natural to take one function and decompose it, i.e. , break it in its
components parts. You can think about functions like to be machines. It accepts a number
'x' like an input, work in it and then give a number 'f(x)' like an output. More complex
machines can be assembled using simpler machines. In the same way you can combine
functions to obtain more complicated functions. In fact, this is what occurs when
functions are composed.
Warm up discussion: [5 minutes]
Provide an introductory diagram of machines that shows how we obtain the composition
of two functions. Prompt students to point out the comparison of how machines operate
to produce the composition of two machines , i.e. the composition of two functions.
f(x)
x
g(f(x))
(g o f)(x) = g(f(x))
g(x)
x
f(g(x))
(f o g)(x) = f(g(x))
1. The Algebra of functions
Find the sum, difference, product, and quotient of functions:
Warm up example or activity:
[5minutes]
Start with a few simple examples of functions, such as f(x) = x + 2 and g(x) = x2, and ask
the students to try to add these two functions. What is f(x) + g(x) ? What about f(x) –
g(x), or f(x)g(x), or f(x)/g(x)? After the students have tried each operation walk them
through the solutions. This is meant to give them an intuitive idea of what is going on.
Formal Concept
[5 minutes]
State the definition of sum, difference, product and quotient of two functions f(x) and
g(x). Stress that the domain of the new functions may be different from the domains of
f(x) and g(x).
Examples:
[10minutes]
1. Given f(x) = x2 +3x-2 and g(x) = x3-3x+5
State the domains of f(x) and g(x)
Find (f+g)(x) and (f-g)(x)
State the domains of (f+g)(x) and (f-g)(x)
Evaluate (f+g)(0) and (f-g)(-1)
2. Given f(x) = 1  x and g(x) = x2
State the domains of f(x) and g(x)
Find (fg)(x) and (f/g)(x)
State the domains of (fg)(x) and (f/g)(x)
Evaluate (fg)(0) and (f/g)(0)
Composition of functions:
Warm up example or activity:
[10 minutes] Give a real life situation. For example, when going shopping at a store in the
mall, you saw an item that worth $100. However, that item is sale for 20% off, and also
every item in that store will have another $20 off. If you want to buy that item, would you
like the cashier to take off $20 first , then 20% or take off 20% first , and then $20 ? Or
you don’t care ?
Prompt the students in such a situation and have them calculate the amount of money
they have to pay and determine which way will cost less money.
Connect the situation with the following functions:
f(x) = x- 20%x = 0 .80x
g(x) = x-20
Formal Concept
[5 minutes] State the definition of the composition of two functions, together with the
notation and its domain. Stress the importance of the order of composition.
Example
[10 minutes]
1. Find ( f  g )( x) and ( g  f )( x) where f(x) and g(x) are given above
Evaluate ( f  g )(100) and ( g  f )(100)
2. Find ( f  g )( x) and ( g  f )( x) and state the domains
a) f(x) = x +1 and g(x) = x3
b) f(x) = x  4 and g(x) = x2
c) f(x) = (1/3)x – 3 and g(x) = 3x+9
Identify a composite function:
Warm up example or activity:
[5 minutes]
Again, start with the examples that were used earlier, f(x) = x + 2 and g(x) = x2. Walk
the students through finding f(g(x)) and g(f(x)). Tell them that what you are doing is
taking the composition of these functions.
Example
[5 minutes]
Find two functions f and g such that ( f  g )( x) = h(x)
h(x) = (x+3)2
1
h(x) =
x5
h(x) = 9  x
Application problems
[8 minutes]
In class exercise
#57 page 194
2. Inverse functions:
Motivation: [2 minutes]
From a “real world” perspective being able to take the inverse of functions enables
engineers to solve many equations necessary for a wide variety of manufacturing and
construction problems. From a point of view of this class being able to take the inverse
of a function will be an important tool in understanding the functions that we are
studying.
Introducing the Inverse of a Function:
Warm up discussion:
[5 minutes]
Put up easy example of a function and its inverse. For example,
and f –1 (x) = x – 1
can be used. Use this easy example to show the following,
f ( f –1 (x) ) = (x – 1) + 1 = x and f –1 ( f (x) ) = (x + 1) –1 = x..
Point out how these two functions “undo” each other.
Warm up example or activity:
[ 5 minutes]
Give the students one or two more examples very similar to the one above and ask them
to find the inverse. For example, use
f (x) = x + 6
f (x) = x/2
While we have not yet defined the inverse of a function formally, the students can use
this experience to get an intuitive grasp on the idea if an inverse function as well as give
them some experience in finding the inverse of a function by trial and error. For many
very simple functions like the one above trial and error may be the easiest way of finding
an inverse.
Formal concept:
[ 3 minutes]
State the definition of the inverse of a function, making sure the students understand that
the domain of f must be equal to the range of f –1 and the range of f must be equal to the
domain of f –1. It may be useful to draw a diagram on the board illustrating this fact.
Also, it is important to take a moment to explain the f –1 notation. f –1 ( x ) does not mean
1 / f ( x ).
Example:
[ 10 minutes]
Provide a more complicated example of a function and its inverse. Ask the students to
take a few minutes to show that each is the inverse of the other. For example,
f ( x)  6 x 3  4
and
g ( x)  3
y4
6
Ask the students to write down f –1 and g –1 to ensure they understand the inverse function
symbol. Once the students have had several minutes to work on the example walk
through it with them, making sure they understand the reasons behind each step.
Finding the Inverse of a Function:
Formal concept:
[ 5 minutes]
Now that the students have had a little bit of experience working with a few examples it
is time to introduce the steps in finding the inverse of a function. It may be best in simply
write the five steps outlined in the text (page 199) on the left side of the blackboard. A
few moments should be taken to point out that not all functions actually have inverses.
Make sure the students understand that it is important to memorize these steps and it is
important that they learn to recognize the cases in which a give function does not have an
inverse.
Example:
[ 10 minutes]
On the right hand side the blackboard, beside the steps for finding the inverse of a
function, put up an example, asking the students to find the inverse of that function. For
example, use something like:
f ( x) 
7x  9
15
After giving the students a couple of minutes to work at the problem do the first step for
them, explaining that what you are doing is exactly what is required in the first step.
Again, give them a few moments to work on the second step. Proceed in this way until
you have done all five steps, explaining each one in detail.
The Graph of the Inverse of a Function:
Warm up example or activity:
[ 3 minutes]
Put on the board the graphs of some function and its inverse. Ask the class to look and
the board and guess at how the two graphs are related.
Formal concept:
[ 2 minutes]
Point out that the graph of the inverse of a function is the reflection of the function across
the y = x line. In other words, if the point ( a , b ) lies of the graph, then so does ( b , a ).
Example:
[ 10 minutes]
Give the students an example, f ( x ) = 5x – 7, and ask them to first find the inverse of
this function, and then graph both the function and the inverse of it. Then ask them to use
their calculators and graph the two functions used earlier in the class,
f ( x)  6 x 3  4
Formal concept:
[ 5 minutes]
and
g ( x)  3
y4
6
Explain the horizontal line test for inverse functions. Tie the horizontal line test for the
inverse of a function in with step three of finding the inverse of a function. If time
permits explain the reasoning behind the horizontal line test.
Follow up assessment
Homework is sufficient.