# Algebra 2: Finding the inverse of a function

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DS is currently enrolled in 2 Algebra 2 classes.  He's taking Algebra 2 through our district's "home and hospital" program, where they send a tutor to work through the curriculum with him, but the tutors have been so erratic that I also signed him up for Algebra 2 through Florida Virtual school.  Our plan is that he'll stay in the Florida Virtual School class until right before the final, and then drop it.

Right now he's doing inverse functions through both programs, which, in theory, should be great right?  But the 2 of them seem to be going about it differently.

Florida Virtual School teaches it like this:

Take an equation like f(x) = 3x - 2

Substitute y for f(x) --- y = 3x-2

Flip the x and y --------x = 3y - 2

Solve for y                  -- y = (x+2)/3

Easy, we have no problem doing that.

But his PS math tutor/teacher wants him to do it some other way that involves "doing the opposite of the steps you'd use to solve the equation, in the opposite order.  He says that on the exam, they'll be looking for him to use this method when he shows his work.  But our school district doesn't have textbooks, so I don't have a worked example, and my kid forgot how to do it. Can someone tell me if I'm doing it right, before I teach him?

I take the equation --- f(x) = 3x-2

If I was solving for a given x, the first thing I'd do would be to multiply the number by three.   Then I'd subtract 2

So I should the inverse operations in the opposite order?  First I should add 2, then I should multiply by 3?

Is that the thinking?  So, he should write out the steps he would use to solve it, and then undo them?

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DS is currently enrolled in 2 Algebra 2 classes.  He's taking Algebra 2 through our district's "home and hospital" program, where they send a tutor to work through the curriculum with him, but the tutors have been so erratic that I also signed him up for Algebra 2 through Florida Virtual school.  Our plan is that he'll stay in the Florida Virtual School class until right before the final, and then drop it.

Right now he's doing inverse functions through both programs, which, in theory, should be great right?  But the 2 of them seem to be going about it differently.

Florida Virtual School teaches it like this:

Take an equation like f(x) = 3x - 2

Substitute y for f(x) --- y = 3x-2

Flip the x and y --------x = 3y - 2

Solve for y                  -- y = (x+2)/3

Easy, we have no problem doing that.

But his PS math tutor/teacher wants him to do it some other way that involves "doing the opposite of the steps you'd use to solve the equation, in the opposite order.  He says that on the exam, they'll be looking for him to use this method when he shows his work.  But our school district doesn't have textbooks, so I don't have a worked example, and my kid forgot how to do it. Can someone tell me if I'm doing it right, before I teach him?

I take the equation --- f(x) = 3x-2

If I was solving for a given x, the first thing I'd do would be to multiply the number by three.   Then I'd subtract 2

So I should the inverse operations in the opposite order?  First I should add 2, then I should multiply by 3?

Is that the thinking?  So, he should write out the steps he would use to solve it, and then undo them?

It is exactly the same procedure.

f(x) = 3x-2

First "undo" the  -2 by adding 2:

f(x)+2= 3x

The "undo" the times 3 by dividing by 3

(f(x)+2)/3=x

So if f(x)=y is the new variable, the inverse function is g(y)=(y+2)/3

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â€¦ Take an equation like f(x) = 3x - 2 â€¦

So I should the inverse operations in the opposite order?  First I should add 2, then I should multiply by 3?â€¦

I think you mean "Add 2, then multiply divide by 3"

So like this: [i will go ahead and substitute y for F(x)]

y = 3x - 2

â€¢ add 2: y + 2 = 3x - 2 + 2

y + 2 = 3x

â€¢ divide by 3: (y + 2)/3 = 3x/3

(y + 2)/3 = x

â€¢ then switch the x and y: (x + 2)/3 = y

which is the same result as your first method.

Best wishes.

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It is exactly the same procedure.

f(x) = 3x-2

First "undo" the  -2 by adding 2:

f(x)+2= 3x

The "undo" the times 3 by dividing by 3

(f(x)+2)/3=x

So if f(x)=y is the new variable, the inverse function is g(y)=(y+2)/3

But what you just did is the same as flipping the variables, right?

It's the "we want you to use a different method" that's tripping me up, I think, and the "reverse the order".

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I think you mean "Add 2, then multiply divide by 3"

So like this: [i will go ahead and substitute y for F(x)]

y = 3x - 2

â€¢ add 2: y + 2 = 3x - 2 + 2

y + 2 = 3x

â€¢ divide by 3: (y + 2)/3 = 3x/3

(y + 2)/3 = x

â€¢ then switch the x and y: (x + 2)/3 = y

which is the same result as your first method.

Best wishes.

Yes, that was a typo.

Teacher said,  "this way will help him really understand the math"

I am not really clear why

flipping the variables then solving for X

is any more or less clear than

solving for y and then flipping the variables

Sorry to be so obtuse, I feel like this is just another "flipping the variables" version.

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But what you just did is the same as flipping the variables, right?

It's the "we want you to use a different method" that's tripping me up, I think, and the "reverse the order".

reversing the order is what you do when you solve for y or when you solve for "f(x).

"We want you to use a different method" smells very much of a teacher who is lacking the full understanding to see that the methods are exactly the same.

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â€¦

Sorry to be so obtuse, I feel like this is just another "flipping the variables" version.

:iagree:  (Not, to the obtuse part; I think the instructions are confusing.)

reversing the order is what you do when you solve for y or when you solve for "f(x).

"We want you to use a different method" smells very much of a teacher who is lacking the full understanding to see that the methods are exactly the same.

:iagree:

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Yes, that was a typo.

Teacher said,  "this way will help him really understand the math"

I am not really clear why

flipping the variables then solving for X

is any more or less clear than

solving for y and then flipping the variables

Sorry to be so obtuse, I feel like this is just another "flipping the variables" version.

The methods are the same. But giving the teacher the benefit of doubt, he/she seems to be asking that the student not flip the variables until the end.

It's like saying, "Here is my equation f(x)=something-about-x. I am going to rearrange my equation so it reads x=something-about-f(x). That shows me how to find x when someone tells me f(x) -- which is what an inverse function means. Aha! So this new equation is my inverse function. Let me give it a new name, g(x), and write it in regular form..."

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