Topic
Inverse Function- Algebraic
Chapter
AA:5.5; HA: 7.2


Goal
Understand the
definition of inverse
relation.
Algebraically find the
inverse relation.
Homework
7.2.1 Bookwork pg 501:2-12
(all)
AA: 5.5; HA: 7.2

Understand how to
graph inverse
relations.
7.2.2 Bookwork
Review on Exponents
AA:5.2

Review the
properties of
exponent such as
power, quotient,
and power of power
properties.
7.3.1 Worksheet
Intro to Logarithmic Functions
AA: 5.6; HA: 7.3

Understand the
definition of
Logarithm.
Understand that the
Logarithm is an
inverse of the
exponentials.
Understand how to
derive a correct
logarithmic
expression for a
given exponential
function and vice
versa.
7.3.2 Worksheet
Inverse Function- Graph
7.1/7.2 Quiz


7.3 Quiz
Properties of Log Functions
AA: 5.7; HA: 7.4

Understand product,
quotient and power
property of log.
7.4.1 Bookwork:
-Page 516-517: 20-28 all
Properties of Log Functions
AA: 5.7; HA: 7.4

Understand the
inverse property of
log.
Use inverse property
to evaluate log.
7.4.2 Bookwork:
-Page 516-517; 519: 11-14,
29-30, 78, 79
Understand how to
use change of base
property of log.
7.4.3 Bookwork:
-Page 518: 51-53,57,59.6063,65

Properties of Log Functions
AA: 5.8; HA 7.4

Exponential Function and Log
Problem
HA: 7.5

Use the properties
of log to solve
problems that
involve exponential
equality.
7.5.1 Worksheet
HA: 7.6

Understand how to
use different
properties of
logarithms to
evaluate natural
logs.
Understand how to
use the formula for
compound interest
in a real-world
situation.
Understand how to
find the half-life of a
given situation.
Understand how to
use half-life to find
the decay-constant.
7.6.1 Bookwork
Page 534:6-10,17-20,24
7.4/7.5 Quiz
Log with base e

Log with base e
HA: 7.6


7.6.2 Bookwork
Page 534-535:
Session 1—Inverse Relations and Functions: Algebraic Approach (7.2)
Goal:
1. Students will understand the definition of inverse relation and algebraically find the inverse.
Objectives:
1. Students will be able to compare the relationship between the relation and its inverse.
2. Students will understand how to find the inverse relation.
Materials:
1. Classroom essentials: Notes, pencils, and calculator
2. Warm-up activity for the day.
Sequence of the Lesson:
1. Reminder on what is a relation/function
2. Lecture about the definition of inverse relation— Function that undoes each other
a. First illustration will be of a vending machine. Vending machine takes the money from you and gives
you an item. Inverse-Vending Machine takes the item from you and gives you back the money.
i. Highlight that the input and output is changed
ii. Highlight that the order of instruction is backward and opposite.
b. Provide an illustration of a function machine to demonstrate the relationship between the output and
input of an inverse function.
i. Input: 3 -> f(x)=x+6 -> output: 9
Input: 9 -> f^-1(x)=x-6 -> output: 3
ii. Possible Misconception
1. The input of an inverse function is not an independent variable
a. Inverse function may be derived from an original function, but I can still put
whatever I want to within its domain.
b. Inverse function is not part of the original function. They are independent from
each other
c. Following from the previous illustration, highlight that the inverse function does the opposite of its
original function
i. F(x)=x+6
1. Instruction: add 6 to the input
ii. F^-1(x)= x-6
1. Instruction: Subtract 6 to the input
iii. Possible Misconception:
1. You only change the operation, but the order stays the same
a. This will be addressed in the next example
d. Multi-Step function
i. F(x)=3x+6
1. Input is multiplied by 3, and then added by 6.
ii. F^-1(x)= (x-6)/3
1. Input is subtracted by 6, then divided by 3
3. Have students work on finding the inverse function mn of 3-4 functions.
4. Transition from the previous activity. Make a remark: to find an inverse relation, you must switch the x and y,
and then solve for y.
Homework:
 7.2.1 Bookwork
Session II—Inverse Function: Graphical Approach (7.2)
Goal:
1. Students will understand how to graph inverse relation.
Objectives:
1. Students will understand why they must switch the x and y value to achieve the ordered pair for the inverse
relation.
2. Students will understand that the graph of inverse relation is achieved when you reflect the original graph over
the y=x line.
Materials:
1. Classroom essentials: Notes, pencils, and calculator
2. Warm-up activity for the day.
Sequence of the Lesson:
1. Brief summary of what we did the previous day.
2. Highlight the point that inverse relation/function switches the input and the output, and connect this idea to the
graph of the inverse relation/function.
a. Example
i. F(x)
X
0
1
2
4
8
y
2
4
5
6
7
1. Graph f(x) by connecting the dots.
ii. F^-1(x)
X
2
4
5
6
7
y
0
1
2
4
8
1. Graph the inverse by connecting the dots.
b. Draw x=y line and make a remark that the graph of inverse relation seems to be a reflection of the f(x)
over x=y.
c. Ask students to graph different given relations to see whether the statement above is true.
d. Possible Misconception:
i. Possible misconception may arise with the label of x and y. Student may think that x-axis
become y-axis.
3. Do an example where you are asked to graph the inverse function to a given function
a. Example
i. F(x)=5x+10
1. First, re-write in terms of y and x
a. Y=5x+10
2. Switch y and x
a. X=5y+10
3. Solve for y
a. ½(X-10)=y
4. Graph ½(x-10)=y
b. Possible Misconception
i. There may be a misconception with knowing how to graph linear equation.
1. E.g not knowing how to find the y-intercept/slope
Homework:
 7.2.2 Bookwork
Session III—Review on Exponents (7.3)
Goal:
1. Students will review the properties of exponent.
Objective:
1. Students will understand the definition of negative exponents.
2. Students will understand the quotient and product property.
3. Students will understand the power of a power property.
Materials:
1. Classroom essentials: Notes, pencils, and calculator.
2. Warm-up activity for the day.
3. Investigation Worksheet.
4. 7.3.1 Worksheet.
Sequence of the Lesson:
1. Talk about the definition of positive exponent, the exponential form, and the expanded form.
a. 4^3=4*4*4.
2. Let students do the part 1 of the Investigation Worksheet.
3. Go over Part 1 with them.
a. Possible Misconception:
i. exercise C (after step 2)
1. Students may not recognize that e has a power of 1.
4. Let students do the part 2 of the investigation worksheet.
5. Go over Part 2 with them
a. Possible misconception:
i. Step 3-C
1. Students may calculate the expression using the calculator.
6. Let Students do the part 3 of the investigation worksheet.
7. Go over part 3 with them
a. Students may have trouble with Step 6.
b. Possible misconception:
i. Eg. 2^3/2^4
1. Students may not know how to come to a correct answer using the quotient property.
8. Let students do the part 4 of the investigation worksheet.
9. Go over Part 4 with them.
Homework:
 7.3.1 Worksheet.
Session IV—Intro to Logarithmic Functions (7.3)
Goal:
1. Students will be introduced to logarithmic functions.
Objectives:
1. Students will understand that Log functions are inverse functions of exponential functions.
2. Students will understand the definition of logarithms.
3. Students will understand basic properties of logarithms.
4. Students will understand how to derive a correct logarithmic expression for a given exponential function and
vice versa.
Materials:
1. Classroom essentials: Notes, pencils, and calculator.
2. Warm-up activity for the day.
3. 7.3.2 Worksheet.
Sequence of the lesson:
1. Introduce logarithmic functions as an inverse function of exponential function.
a. F(x)= b^x=a
i. X is an input, a is an output
b. F^-1(a)= logb (a)= x
i. Logb is a label
ii. A is an input
iii. X is an output
2. Introduce the definition of logarithmic function.
a. If 𝑏 𝑥 = 𝑎, then log 𝑏 𝑎 = 𝑥
b. Go away from the notion that a logarithmic function is an inverse function of an exponential function.
3. Introduce 𝑊𝐸𝐺𝑂𝑏 𝑎 = 𝑥
a. What Exponent Goes On b to give me a? The answer is x
4. Introduce the common logarithm.
a. If there is no base in LOG, we are to assume that it is base 10.
5. Provide examples that translate an exponential function to a logarithmic function, and vice versa.
a. Example
Exponential Function
WEGO
Log
3
What
Exponent
Goes
On
2
to
Log
2 =8
28=3
give me 8? 3
WEGO28=3
i. Possible Misconception
6. Provide examples that evaluate simple logarithms
a. Log4 (1/4)
i. Emphasize the WEGO question
1. What Exponent Goes On 4 to give me ¼?
ii. Re-write it into an exponential form
1. 4x = ¼
iii. Unify the base
1. 4x= 4-1
2. X has to be -1
3. So log4 (1/4) = -1
b. Possible Misconception
i. Student may struggle with re-writing the logarithm to an exponential form.
ii. Student may struggle with unifying the base, due to some unfamiliarity with negative exponents.
Homework:
 7.3.2 Worksheet.
Session V—Properties of Log Functions (7.4)
Goal:
1. Students will be introduced to the Properties of Log
Objectives:
1. Students will be exposed to the proof of Product and the power property of log
2. Students will learn how to use Product, quotient, and power property of log to expand and simplify logarithmic
expression.
Materials:
1. Classroom essentials: Notes, pencils, and calculator.
2. Warm-up activity for the day.
Sequence of the Lesson:
1. Start the class with a lecture on the product property.
a. You will prove the product property.
b. Provide an example of the product property.
i. Log312= log33*4= log33 + log34
ii. Possible Misconception:
1. Student may get caught up in the choice of the factors. This may lead students to
believe that only specific factors (i.e factors that are identical to the base) are subjected
to the product rule.
2. Student may extend this property of product of logarithms. (eg log3 * log4 = log 12)
c. Have students work on few problems that involve expanding/simplifying logarithms using the product
property.
2. Introduce the quotient property (state it, and don’t prove it).
a. Provide an example of quotient property
i. Log5 (16/2) = log516- log52
ii. Possible misconception:
1. Similar to the product property, student may extend or confuse this property to the
quotient of logarithms.
b. Have students work on few problems that involve expanding/simplifying logarithms using the quotient
property.
3. Introduce the power property.
a. Quickly prove the power property
b. Provide an example of the power property.
i. Log 52 = 2*log 5
ii. Possible misconception:
1. Student may extend this property to the power of logarithms.
2. Student may interpret the power sign to the whole logarithm, when it on applies to the
input of the log.
4. Have students work on few problems that involve expanding/simplifying logarithms using all the properties.
Homework:
 7.4.1 Bookwork: Page 516-517: 20-28 all
Section VI—Properties of Log Functions (7.4)
Goal:
1. Students will be introduced to the inverse properties of logarithms and exponents.
Objectives:
1. Students will understand the inverse properties of logarithms. (logbbp=p)
2. Students will understand the inverse properties of exponents.
Materials:
1. Classroom essentials: Notes, pencils, and calculator.
2. Warm-up activity for the day.
Sequence of the Lesson:
1. Start with the lecture on inverse properties of logarithms
a. Provide an example
b. log 8 83𝑥+1 = 3𝑥 + 1
i. Possible misconception
1. Student may not know what to do with +1.
2. Student may say prematurely use the same base property and conclude that log883x+1
=13x+1
ii. Try to stress the point that log8x is the inverse of 83x+1
2. Lecture on inverse properties of exponents.
a. Provide an example
b. 2log2 27 = 27
i. Possible misconception
1. 27 is the input of log2. Hence, it is part of the exponent. Student may be confused on
why 27 became a base.
2. It is important to stress that 27 is an input of a log function. Exponential with base 2 is
the inverse. Hence, 2log2 27 will output the input of log227
a. ….This gets too technical… not sure if this is the best explanation.
3. Let students do a couple of problems on their own.
a. Go over the problems with them afterward.
Homework:
 7.4.2 Bookwork: Page 515-517;519: 11-14.29-30,78,79 all
Section VII—Properties of Log Functions (7.4)
Goal:
1. Students will be introduced to the change of base formula.
Objectives:
1. Students will use the change of base formula to evaluate the logarithmic expression.
2. Students will use the change of base formula to graph the logarithmic expression.
Materials:
1. Classroom essentials: Notes, pencils, and calculator.
2. Graphing calculators.
3. Warm-up activity for the day.
Sequence of the Lesson:
1. Lecture on Change of base formula.
2. Provide an example where you can use the change of base formula to evaluate the log expression.
log 8
log 23
2
2
3
a. log 4 8 = log 2 4 = log2 23 = 2
b. Possible Misconception:
i. Students may be confused on why I chose to change to base 2.
ii. Students may attempt to use the quotient property.
1.
log2 8
log2 4
= log 2 4
3. Provide an example where you can use change of base formula to change the log expression to log base 10 and
graph them using the graphing calculator.
log 𝑥
a. 𝑦 = 4 log 3 𝑥 = 4(log 3).
i. Possible misconception
1. Student may divide the log out and graph 4x
Homework:

7.4.3 Bookwork: Page 518: 51-53,57,59.60-63,65
Session VIII—Exponential Function and Log Problem (7.5).
Goal:
1. Students will learn how to solve exponential and logarithmic expressions.
Objectives:
1. Students will learn how to solve exponential equations by unifying the base.
2. Students will learn how to solve exponential equations by logarithms.
3. Students will learn how to solve logarithmic equations by using properties of logarithms.
Sequence of the Lesson:
1. Lecture on solving exponential equations.
a. Provide an example where you have to unify the base of the equation.
i. Example
1. 8𝑥 = 2𝑥+6
(23 )𝑥 = 2𝑥+6
23𝑥 = 2𝑥+6
3𝑥 = 𝑥 + 6
𝑥=3
2. Possible misconception
a. (23 )𝑥 = 2𝑥+6 + 1 Students may equate the exponents in such situations.
b. Have students to work on a set of problems.
i. Go over them.
2. Lecture on solving exponential equations where you cannot unify the base.
a. Provide an example
i. Example
1. 5𝑥−2 = 200
log 5𝑥−2 = log 200
(𝑥 − 2) log 5 = log 200
𝑥 − 2 = log 200 / log 5
log 200
𝑥=
+2
log 5
2. Possible Misconception:
a. Student may conclude that you have to use log base 10.
b. Have students to work on a set of problems.
i. Go over them.
3. Lecture on solving logarithmic equations by using properties of logarithms.
a. Provide an example.
b. Have students to work on a set of problems.
Homework:
 7.5.1 Worksheet.
Session IX—Log with base e (7.6)
Goal:
1. Students will be introduced to natural logarithms and its applications.
Objectives:
2. Students will simplify natural logs using various properties of logs.
3. Students will use A=Pert formula to solve real-life situation.
Sequence of the Lesson:
1. Lecture on Natural Log.
a. Provide examples where you simplify natural log using various properties.
5
i. 𝑒 5ln 𝑥 = 𝑒 𝑙𝑛𝑥 = 𝑥 5
ii. Possible misconception:
1. Student may not comprehend that ln is logarithm of base e.
b. Have students to work on a set of problems.
2. Introduce A=Pert
a. Provide examples where you use the peRT formula.
i. Example
1. What is the total amount for an investment of $1000 invested at 5 % for 10 years
compounded continuously?
2. Possible Misconception:
a. Student many not able to distinguish between continuously compounded
interest versus discreetly compounded interest.
b. Student may say r = 1.05 instead of r = 0.05
Homework:

7.6.1 Bookwork –Page 534: 6,8,9,10,11,17-20,24.
Session X—Log with base e (7.6)
Goal:
1. Students will utilize the concept of half-life to solve given radioactive-decay problems.
Objectives:
1. Students will understand how to use the radioactive decay formula.
2. Students will understand how to use half-life to find the decay-constant.
Sequence of the Lesson:
1. Lecture on radioactive decay formula and half-life
a. Provide an example where students have to find the decay constant given a value for the half-life
i. A paleontologist uncovers a fossil of a saber-toothed cat in California. He analyzes the fossil and
concludes that the specimen contains 15% of its original carbon 14. Carbon-14 has a half-life of
5730 years. Use carbon-14 dating to determine the age of the fossil.
1. Possible misconception
a. This example has multiple steps. Student may need to be directed such that
they know what they are looking for.
b. Student may not understand what they are exactly looking for.
c. Student may interpret 15 % as the decay constant.
ii. Determine how long it will take for 650 mg of a sample of chromium-51, which has a half-life of
about 28 days, to decay to 250 mg.
1. Possible misconception
a. This example has multiple steps. Student may need to be directed.
b. Student may try to find the decay constant by (650/250)/28 days
c. Student may not know what they are looking for. They may stop at finding the
decay constant.
2. Have students work on a set of problems. Go over them as a class afterward. If there are any errors made by a
student, do an error analysis.
Homework:
 7.6.2 Bookwork—page 534-535: 12, 21,22, 30.