Discussion

Define the symbol * by the following equa­tion: x* = 1 – x, for all non-negative x.  If , then x =
(A)1/2
(B)...
(C)...
(D)...
(E)...
(F)...
*This question is included in Nova Press: Problem Solving Diagnostic Test, question #30

The solution is

Posted: 11/16/2011 15:15
I don't understand what * is supposed to mean. Is it supposed to be a power function? I don't understand the substitution in the solution.
Posted: 11/17/2011 11:39
Hi Angie,

The * is an operator.
So whenever you see x*, you have to sub in 1 - x. (As the problem says, x will always be non-negative for this operator.)

So if you have 6*, that's equal to 1 - 6., which is -5.

Now for the actual problem:

As you can see, the left-hand side of the problem has two *'s. Let's simplify that side by take each * on one at a time.

((1 - x)*)* =
((1 - (1 - x))*

Let's clean it up a bit:
((1 - 1 + x))* =
((x))* =

You can see here we've gotten rid of one of the *'s. Now for the other:
x* =
1 - x

So the left-hand side simplifies to 1 - x.

Now for the right-hand side:
(1 - x)* =
1 - (1 - x) =
1 - 1 + x =
x

Let's put 'em together (left-hand side & right-hand side):
1- x = x
1 = 2x
1/2 = x

Hope that helps.
Posted: 11/05/2013 04:32
Hi Brendan,

I'm a bit confused on how you got from step one to two.
((1 - x)*)* =
((1 - (1 - x))*
Since the (1-x) is in parentheses how do you sub (1-x)* for 1-(1-x)?
I don't see how you can get x* derived if the * is just an operator.
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Contributor
Posted: 11/05/2013 12:25
You make the substitution based on the definition of the operator, * (or "raised star"), given in the beginning of the problem: that is, x* = 1-x. We can use parentheses, if it makes it clearer: (x)* = (1-x). Given any expression with a * in it, to translate the * operator (note this is not the multiplication symbol in this problem) to our normal mathematical operations, you use the definition: subtract the value to the left of the * from 1. Therefore, (1-x)* = 1-(1-x).

Again, this problem is made more confusing because the "raised star" operator looks much like the "asterisk" multiplication symbol. But it is not. Hope that clarifies it.
| Edit
Posted: 01/02/2014 05:56
Thank you
Posted: 11/17/2011 11:46
That was quick. I just downloaded this app and was trying the free sample questions. This one got me stuck. Thanks for the explanation. I missed that * is an "operator". I am liking this discussion board already.
Posted: 11/23/2011 03:27
I don't understand the 2nd step lhs
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Contributor
Posted: 11/23/2011 12:51
The second step is simplifying the expression using the commutative property of addition. This property means that numbers can be added in any order and the sum will be the same.

((1 - (1 - x))*
((1 - 1 + x))* = (distribute negative sign and then remove inner parentheses)
((x))* = (subtract 1 from 1)
x* (remove all parentheses)

It's always a good idea for the math section of these tests to be rock solid on the order of operations: PEMDAS (you may have learned this acronym with the mnemonic "Please Excuse My Dear Aunt Sally). The order of operations is Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Of course, multiplication and division are interchangeable, as are addition and subtraction, but it is helpful to have a go to order for every algebraic expression.

In addition, the newly defined function (or "made-up function" as I like to call it) is a common problem type, one that every test taker should, and can, master. This particular problem is especially tricky because the made-up function (*, or "asterisk") looks an awful lot like the multiplication symbol.
Posted: 12/20/2011 20:10
Why doesn't the * cancel out on both sides to leave you with (1-x)* = 1-x
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Contributor
Posted: 12/21/2011 08:53
The symbol, *, is a function operator so it cannot be cancelled out. Similarly, you cannot cancel out the addition symbol, +, in an equation. Again, the test makers made this whole problem potentially more confusing because the new symbol looks like the multiplication operator. More often, however, they use other symbols, such as diamonds, boxes, and the like.

In these types of problems, where the test maker introduces a new function operator, the only way to remove it so that you are left with the "normal" operations (addition, subtraction, multiplication, division, etc.) is to substitute the new operator with it's definition.

The definition of the new operator, *, basically says this: "take everything on the left of the * and subtract it from 1."

In your question, Bryan, you ask about (1-x)*. Everything to the left of the * is (1-x), so,using the definition above, we subtract this from 1:

1-(1-x)

The * is now removed and we can do our normal arithmetic on the result:

1-(1-x )= 1-1+x = x

Hope that answers your question!
Posted: 02/21/2013 19:45
Looks too much like it's supposed to be a power, like (1-x)* = (1-x)^(1-x)

Probably should find a way to make that more clear.
Posted: 04/22/2014 06:52
But, for example:
(X-1)^2= (x-1).(x-1) not equal to:
(X^2-1^2), which is what you guys are impliying?????
Posted: 04/28/2014 16:49
Yitzak, please read the problem carefully. * defines a function of x as 1-x. It does not mean raising to the power in this problem. This problem has also been explained many times by different contributors. Please follow the thread.
Posted: 04/22/2014 06:52
But, for example:
(X-1)^2= (x-1).(x-1) not equal to:
(X^2-1^2), which is what you guys are impliying?????

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