##
* *Reflection: Developing a Conceptual Understanding
Finding the Inverse of a Function Day 1 of 2 - Section 2: Discovering the definition of inverse functions

I taught this lesson during the block day or our weekly schedule. We have one day a week that the class meets for approximately 90 minutes. This means that I have 2 of my classes one day and then the 3rd class the next day.

During the first day of instruction I noticed the student were saying that 2 functions were inverses when f(g(x))=g(f(x)). They did not understand that the result would always give the original domain value. After seeing this in the first class I started thinking about how I could fix this misconception. The same confusion came up in the second class. At this point I had the students evaluate the compositions for x=2. The students saw how the answer came out 2 so they added x.

After thinking more about this issue I decided that the students probably did not really understand that the answer will always come out with the domain value. I decided to expand on the adjustment for the last class. For this class I gave each group a different domain value. The students worked and then shared their answers.

As students worked I heard one comment that the definition said the answer will be x so that means it should be 2 since we were given x=2. Do both answers come out as 2? This was great problem solving and understanding for that group.

I also have a group of students whose algebra skills are weak. I gave this group x=0 so that they could complete the problem quickly. Even with this the group found the 2 answers were not equal. The other groups helped the students find the algebra mistake and corrected their answers.

I plan to use this new idea next year to make sure students really understand the definition of inverses.

*Developing a Conceptual Understanding: Adjustment in instruction*

# Finding the Inverse of a Function Day 1 of 2

Lesson 10 of 15

## Objective: SWBAT find the inverse of a function when the function is represented as a table, graph or equation.

#### Bell Work

*10 min*

The class starts with students reviewing composition of functions. This problem along with yesterday's closure will help introduce the definition of an inverse function.

The closure problem from yesterday gave students an opportunity to see that inverse functions have a composition that comes out to x. The bell work is not obvious since most students do not see that these 2 linear functions undo each other.

After students have worked for a few minutes, students share their process and result from today's bell work. I choose students that have the correct answer but are not confident to share their results.

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After students share their results from the Bell Work, I comment "The answers are the same."Then, I ask

- Will f(g(x)) always be equal to g(f(x))?
- Have we seen problems where they are not equal?
- Have we seen another pair of functions that came out with the same result?

When the students comment that the closure from yesterday had the same result, I ask if anyone remembers the functions.The functions are written on the board and I then ask if there is any relationship between these functions.

Some students tell me they are opposites of each other. I ask what is meant by opposite. Many times, students say "opposite" when the mean "the inverse of." After students say that square root of x undoes x^2, I say "Oh, I thought you meant the sign was different." I then explain that the word **opposite** in mathematics is talking about direction, as in the opposite on a number line. The opposite of 3 is negative 3 since negative 3 is on the opposite side of 0. When we undo an operation we call this an inverse.

I also share the mathematical definition of opposite and inverse. I use the 2 different sites for the definitions: Mathgoodies.com and Mathwords.com. These sites offer definitions that my students find easy to understand. As we discuss the definition, I explain to my students why I will correct terminology when necessary: being precise when we communicate is important for understanding ideas (**MP 6**). So, in the case of f(x)=x^2 and g(x)=sqrt(x) we say the functions are inverses.

I now go back to the sets of functions we have been discussing. I remind students that the compositions from yesterday's functions (x^2 and sqrt(x)) simplified to x and we know that these two functions are inverse operations. I now go to the bell work problem and ask what happened when the compositions were simplified. If my students observe that the composition simplified to the value, x, I will say, "What can you hypothesize about these 2 functions? Can you write a general hypothesis about what will happen when two inverse functions are composed?" I take this approach, because I have found when students discover a definition and write out what they conclude, the students will remember the definition better than if I just give students the definition.

After the class writes a definition I share the definition of inverse from the book. I give students a couple minutes to read the definition on their own. I then ask if there are any words they do not know. I underline the word denoted since this is a word most students do not know. I ask, "By considering the context of the word, what do you think denoted meant?" I then look at the beginning of the definition and ask if this is similar to what the class wrote.

I discuss the notation we will use this notation as we find inverses. This notation is used when the students work with inverse trigonometric functions seeing how f^(-1) is the same as sin^(-1) helps students with the inverse trigonometric functions.

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I now have a student read the last paragraph of the definition. I ask:

- What does this paragraph mean?
- If 3 is in the domain of f(x) what is it for f^(-1)(x)?

I give the students the bell work problems but I have changed g(x) to be written as f^(-1). I ask how I can just change the name of the function. Students need to recognize that we proved these 2 functions were inverses in the bell work.

Students pick some x-values for each function from the bell work and evaluate each function. I make sure the students try the values -1, 1, and 2 for f(x) and 2, -4 and 5 for the inverse. When students see how the ordered pairs are switched they start to see what is meant in the definition.

Now that students have seen how the definition gives a method for finding an inverse I give students a table and graph. Students discuss with their groups how to find the inverse. After a few minutes students share their answers.

Some of the students may bring up how the graphs seem to reflect. I will ask the students to explain what they mean. Some will immediately say the graphs reflect across the y=x line or the students say the diagonal. I graph y=x on the graph so that all students can see the reflection. I then say can we make a hypothesis about the graph of a function and its inverse. We write how the graphs reflect across the y=x.

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As I move to have students find the equation of an inverse is make the following comment, "if we want to find the inverse when we have a table we switch the domain and range. When we find the inverse on a graph, we find points and switch the x and y coordinate. So how can we use this idea to find the equation for an inverse when we know the equation of the original function?" Some students make the comment that we need to switch the x and y. I ask students to clarify what is meant by switching the x and y.

To help students learn to read examples from the textbook. I share example 6 from the text. We move through the example and analyze each step. Some of the questions I ask include:

- Why did the author replace f(x) with y?
- Why did they multiply by 2 first?
- Did anyone want to move the 5 and then divide by -3? Will you get the same result?
- Why did the author exchange the y with the inverse notation?
- Is the y of the inverse the same as the y of the original function?

#### Resources

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#### Closure

*5 min*

As class ends I give students 3 problems to find the inverse equation. At this point we have not discussed one to one this will be where we start tomorrow. The problems include a simple linear function, then move to a quadratic that we will need to limit the domain on tomorrow and the final problem is a rational function.

I am expecting students to struggle with the algebra on the last problem. I will work individually with students on this problem and may do a problem similar to the problem 3 as an example tomorrow.

#### Resources

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Interval Notation
- LESSON 2: Evaluating Piecewise Defined Functions
- LESSON 3: Writing Piecewise Functions
- LESSON 4: Graphing Piecewise Defined Functions
- LESSON 5: Function Notation
- LESSON 6: Operations of Functions
- LESSON 7: Presentation on Functions Operations
- LESSON 8: Composition of Functions, Day 1 of 2
- LESSON 9: Composition of Functions, Day 2 of 2
- LESSON 10: Finding the Inverse of a Function Day 1 of 2
- LESSON 11: Finding the Inverse of a Function Day 2 of 2
- LESSON 12: Transforming Functions Day 1 of 2
- LESSON 13: Transforming Functions Day 2 of 2
- LESSON 14: Review for Assessment
- LESSON 15: Assessment