What is the value of |f(x)| - |g(x)| + |f(g(x)| ?

Question

Let f(a) = a - 5
 g(b) = 5 - b.

A. |x - 10|
B. 3x + 10
C. |x|
D. |x - 5|
E. x

What is wrong with  E

Thanks & Regards
Vinni

Vips0000 · Accepted Answer

From question: |f(x)| = |x-5| |g(x)| =|5-x| |f(g(x)| = |f(5-x)| = |5-x-5| =|-x| Now: |f(x)| - |g(x)| + |f(g(x)| = |x-5| -|5-x|+|-x| =|x-5|- |x-5|+|x| =|x| Ans C. It can not be E because x <> |x| for any negative value of x. Hope it helps..

Bunuel · Answer

|-x|=|x| for all x's. Now, if x<0, then |-x|=|x|=-x but if x>=0, then |-x|=|x|=x.

Hope it's clear.

vinnik · Answer

Well, I haven't understood completely.
All i know from my knowledge that |a - b| = |b - a|
So, as you have explained above |x-5|- |x-5| = 0
Now we are left with |-x|
According to what i have read from the books, any absolute value whether negative or positive will come out as positive. For eg. |-5| = 5.
This is the exact reason i selected E as the answer.

I must be missing one of the concepts. Can you please elaborate more on it. I didn't completely understand this statement " It can not be E because x <> |x| for any negative value of x. "

Thanks & Regards
Vinni

Vips0000 · Answer

Two points that you are confused with: |-x| = |x| This is exactly same thing as you have mentioned: "All i know from my knowledge that |a - b| = |b - a|" Now second point, x <> |x| for any negative number x. Well, take for example x =-5 in this case, x= -5 and |x|=5 Are these 5 and -5 equal? no. That is x <> |x| for any negative number x. What you are confusing this is with |x|=|-x| Hope it is clear.

greatps24 · Answer

how i solved this ques:

|f(x)| - |g(x)| = 0.  Because, f(x) and g(x) both represents distance between x & 5.

Therefore, we have to solve only this |f(g(x)| => |f(5-x)|= |5-x-5| = |x|

2013gmat · Answer

I am confused as in the value of |-x| will always be x so why are we choosing |x| ?

greatps24 · Answer

Kudos Bunuel. Thanks for clarifying

One point: what if in answer choices we also have |-x| (and |x|)?

Bunuel · Answer

|-x| and |x| are equal, thus we cannot have both of them among answer choices. Consider this, can we have both 4 and 2^2 among answer choices?

2013gmat · Answer

Kudos Bunuel. Cheers

mani6389 · Answer

Got a doubt...

f(g(x)| = |f(5-x)| = |5-x-5| =|-x|
Why (5-x) is considered without Mod sign. Ideally it should have been |f(|5-x|)| = |(|5-x|)-5|
If we simplify this we get two options |x-10| and |x|.
Why this is not correct ??

Bunuel · Answer

|f(g(x))| has only one modulus.

g(x)=5-x, thus |f(g(x))| = |f(5-x)|.

Next, since f(5-x) = 5-x-5=-x, then |f(5-x)| = |-x| = |x|.

Hope it's clear.

mehasingh · Answer

Let f(a) = a - 5
g(b) = 5 - b.

What is the value of |f(x)| - |g(x)| + |f(g(x)| ?

A. |x - 10|
B. 3x + 10
C. |x|
D. |x - 5|
E. x

hi in the above question it is asking the value of the equation 
|f(x)| - |g(x)| + |f(g(x)|....

when we put in the values appropriately:

|x-5|-|5-x|+|5-x-5|
=|-x|

mod of -x=x

i understand the piece tht

when we have the value as -x and we take modulous then it gives the values 
as positive values of x.

|-x| is not equal to x

but here it is asking us the eventual result of the equation
so when we get |-x| we get the final result, it is positive values of x.

KarishmaB · Answer

You are assuming that x is positive.

|-x| = |x| in any case. x may be positive or negative. Take examples:
Say x = 5, |-5| = |5| = 5
Say x = -5, |-(-5)| = |-5| = 5
Hence (C) is correct.

But to remove the mod, you must know the sign of x.

By definition,
|x| = x when x is positive 
|x| = -x when x is negative

|-x| = |x| = x only when x is positive
If x is negative, say x = -1, 
|-x| = |-(-1)| = 1 which is not the same as x.

Hence  |-x| 
eq x  when x is negative.
Since we have no information on the sign of x, we cannot say that |-x| = x.

kuttingchai · Answer

Can any1 explain why

= |x-5| -|5-x|+|-x|

=|x-5|- |x-5|+|x|

|5-x| = |x-5| ???

Thank you

Bunuel · Answer

Yes, |5-x| = |x-5|, because both represent the distance between x and 5.

fozzzy · Answer

one more question if we were given X is positive or negative then the answer would be E?

Zarrolou · Answer

Not sure of what you mean, but the answer is  C   for every value of x  (+ve, -ve or =0)

fozzzy · Answer

As per Vips000 solution after you solve we get  =|x|

now if we were given an additional constraint that X is positive then we can solve for the modulus right?

Zarrolou · Answer

If we know that x is positive then we can solve the modulus and obtain  |x|=x , yes you're right.

But be careful, in this case both c and e would be correct

C).  |x|  =>  x

E).  x 
They are the same.

Hope it's clear

What is the value of |f(x)| - |g(x)| + |f(g(x)| ?

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What is the value of |f(x)| - |g(x)| + |f(g(x)| ?

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