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Inverse Statements


Date: 08/20/98 at 20:02:40
From: Axel Ishmael
Subject: Half empty or half full?

I have been studying inverses. It is my understanding that the same 
situation can be described using an inverse statement.

For example, say that there are 3 + 4 times as many girls as boys. Is 
this the same as saying there are -3 + 1/4 as many boys as girls?

But I still don't quite understand why the inverse of (3 + 4X) must 
equal -(3 + 4X) which is -3 - 4X.

Do you see that the definition of inverse seems to be in conflict here?  
Would it be correct to define say that there are -3 - 4times as many 
boys?

Does -(3 + 4X) also equal (1/3 + 1/4X) because the negative sign 
means inverse?

I am trying to describe the same situations using inverses.

Thank you for all your help.


Date: 08/21/98 at 16:13:00
From: Doctor Peterson
Subject: Re: Half empty or half full?

Hi, Axel. 

I think it might help if we take out some of the words and write a 
complete equation to clarify what you are saying. Talking about 
"3 + 4times as many girls as boys" is confusing, because it's not 
really proper grammar in either English or math.

Let's say that there are X boys and Y girls. Now I think you are 
saying that

    Y = 3 + 4X

Now you want to invert this statement, to say how many boys there are 
compared to girls, rather than girls compared to boys. That means we 
want to solve for X in terms of Y.

You're right that we want to use inverses, and that -3 is the additive 
inverse of +3 and 1/4 is the multiplicative inverse of 4. Notice that 
there are two kinds of inverses, so the potential for confusion is 
obvious if we don't discuss them in the right context. A better way to 
describe this is that subtracting 3 is the inverse of adding 3, and 
dividing by 4 is the inverse of multiplying by 4.

The other trouble you run into is that if you want to invert a whole 
expression, you have to invert its parts in the right order. It's as 
if I wanted to take off my clothes, and I tried to remove my shirt 
first, then my sweater, then my coat. It wouldn't work. I'd get all 
tangled up, just the way you did.

An algebraic expression comes in layers - we call this the order of 
operations. In your expression, we first multiply 4 times X, then add 
3 to the result. Now we want to peel off those layers one by one and 
wrap them (inside out - inverting them!) around the other side of the 
equation. So since the "outside layer" is adding 3, we invert it by 
subtracting three from both sides:

    Y - 3 = 4X

Now to unwrap the next layer, multiplication by 4, we divide each side 
by 4:

    Y - 3
    ----- = X
      4

So now we have inverted your expression, and the result is not 
-3 + 1/4Y, or -3 - 4Y, or 1/3 + 1/4Y, but (Y - 3)/4. Order makes a big 
difference.

Now, I don't know whether you are learning about inverse functions, 
but if I want to say this in terms of functions, I can define a 
function:

    f(x) = 3 + 4x

and what we have found out is that the inverse of this function is:

    f^-1(y) = (y - 3)/4

This is the function that undoes what f did. Function f dressed x, and
f^-1 undresses it. Function f changed the number of boys into the 
number of girls, and f^-1 changes the number of girls back into the 
number of boys.

I hope this helps. You're obviously thinking hard and enjoying it, and 
keeping things in order should make your thinking more satisfying.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Functions

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