Lesson Title: F.BF.4 Inverse functions vs. Non-functions and restricting domain Course: _Algebra 1 Unit 5
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
F.BF.4 Find inverse functions. Focus on linear functions, but consider simple situations where the domain of the
function must be restricted in order for the inverse to exist, such as f(x) = x2, x > 0.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the
inverse. For example, f(x) = 2x3 or f(x) = (x + 1)/(x – 1) for x not equal to 1.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Graph the following set of data.
x
-1
0
1
2
3
f(x)
-5
-4
-3
-2
-1
Does this relation represent a function?
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow? List the questions.




How could you graph the inverse of a function, given
only the graph?
How might you graph an inverse of a function, given a
table of values?
How might you figure out the inverse of a function
from an equation?
Can you come up with a way to figure out if an
inverse is a function directly from the graph of the
original function?
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson?
1. After making sure everyone has the data graphed correctly, have students explain the definition of a function and
how that definition relates to an equation, a table of values, and a graph.
2. Have the students write down directions to get to the front office. Have one student share his or her directions.
Tell them to be as specific as possible. Write these directions on the board.
3. Have students write directions to get back to the classroom the same way. Have them show that the steps you
take to get back from the office are the reversed steps from what you did to get there.
4. Explain that a function works in a similar fashion. With input value, x, the function gives the direction and the
output of y. Use the table of values from the warm-up to illustrate how functions work. “If the input, x, is 2,
then the output, y, is -2”, etc. Say, “As we did with our directions, we want to ‘get back.”’ Our directions are
what we call the Inverse. The inverse reverses the inputs and the outputs. With the inverse on our table of data,
if the input is -4, what is the output?” Give students a chance to answer. Use more examples to help students
solidify the understanding of inverse. Have students graph the points of the inverse function in a different color.
Ask, “What do you notice about the graph of the points of the inverse?” Have the students make conjectures
about what is happening graphically with the inverse.
5. Pair students and write on the board: Multiply the input by 2, and then add 4. Have the students write an
equation for this function where x is the input and y is the output. Ask, “If the function is to multiply by 2 and
then add 4, what would the inverse be? Remember that the inverse is to “un-do” the directions given.” Have
them write the equation for the inverse with x as the input and y as the output. Ask the students to take a
moment to think about how the function and the inverse of the function are related. Have students discuss with a
partner and then solve the inverse function for x. Encourage students to see that an alternate definition for
inverse functions is to switch the x and the y. Have the students graph the function and its inverse on a different
graph in two different colors. Have the students discuss with their partners their previous conjectures about the
graph of an inverse function. Do these conjectures hold true with this linear function?
6. Have the groups construct a hypothesis and then graph the function f (x)  x 2 in one color. Groups should be
able to figure out that the inverse will be the square root of x, but they should create a table of values to graph the
inverse. Have groups discuss what they see in the graph and assess whether or not it conforms to their original
conjecture. Students should see that an inverse reflects the original function over the line y=x. The appropriate

HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible
mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: F.BF.4 Inverse functions vs. Non-functions and restricting domain Course: _Algebra 1 Unit 5
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
notation for an inverse function is f (x) 1 .
7. Groups should recognize that the inverse of f (x)  x 2 is not a function. As a class, discuss how the graph
would have to be modified to create a function. While students may express that half of the graph would need to
be removed, have them think about the inverse as a square root and what values of the graph would not make

sense. Students should build to the idea that the inverse of f (x)  x 2 , where x > 0 is f (x)1  x .

Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
 (and conceptual) understanding.

observer) that your students have developed a deepened
Do the students understand the concept of inverse functions as an equation, graph or set of data?
Can the students find an inverse given multiple means?
Do students recognize that, although a relation may be a function, its inverse may not?
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Define relation, function and inverse operation.
Students may have different ideas on the best way to get to the office. Have students use their own specific
directions to find the way back.
When writing the equation of the inverse, students may want to follow the order of operations and divide by 2 then
subtract 4. Make sure they realize that you are reversing the steps in order. They also may not realize that the
division by 2 applies to both the -4 and the x.
f 1 is sometimes mistaken as
1
. Make sure students realize that the superscript -1 notes inverse function in this
f
context.

Resources: What materials or resources are essential

for students to successfully
complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Graph paper with multiple graphs
Graphing calculator
Colored pencils
Find and graph the inverse of the function for the area of a
circle and explain any constraints necessary on the
domain.
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use to
reflect on the effectiveness of this lesson?
Can students identify inverses from equations, graphs and data points?
Do students recognize that a function is reflected over the line
Do students recognize that a function is reflected over the line y = x?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.