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Function and square root strategy

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Function and square root strategy [#permalink] New post 27 Oct 2012, 04:14
Hi,

I have a conceptual question. It is very easy, but i am confused. This is from MGMAT, 4th edition, Equations inequalities and VIC's book.
Chapter no. 5 (functions strategy) Pg no. 73

The question is :-

If f(x) = x^3 + √x and g(x) = 4x - 3, What is f(g(3)) ?
Solution as given in the book:-
g(3) = 4(3) - 3 = 9

f(g(3)) = f(9) = 9^3 + √9 = 729 + 3 = 732

My doubt :- √9 - the square root of 9 must have two solutions, one is 3 and another is -3 (3,-3) which give two answers.
Why are we sticking to the positive solution of √9.

This strategy is used in couple of questions (Q4 and Q5) same chapter.

There is a similar question in the same chapter, pg no. 79, Q2
If g(x) = 3x + √x , What is the value of g(d^2 + 6d + 9) ?

Now here are two solutions:-
g(d^2 + 6d + 9) = 3(d^2 + 6d + 9) + √(d+3)^2
This √(d+3)^2 gives two solutions as given in the answer explanation.

+(d+3) and -(d+3) which give two answers.

One more question in O.G-12, DS, Q36.

For all integers n, the function f is defined by f(n) = a^n, where a is constant. What is the value of f(1) ?
(1) f(2) = 100
(2) f(3) = -1000.

My focus is on the (1) which clearly says that √100 has two solutions.

I really apologize if i have given a very long explanation, but i am really confused by the contradicting explanations. I hope i have clearly explained my doubt.

Looking forward to your replies.

Thanks & regards
Vinni
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Re: Function and square root strategy [#permalink] New post 27 Oct 2012, 15:48
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"Roots have only one solution." MGMAT Number Properties Guide (4th) p.77. Read this section, it explains it very well imho (if you still have questions or don't have the Number Properties Guide, just tell me)
This is what you need to consider when you're solving the problem with f(x) and g(x)


On the other hand, if you have an expression with an even exponent and take the root you do not know whether the result is positive or negative, i.e. you have two possible solutions. It's maybe clearer when you take an expression such as √x^2. This equals |x|, which yields a negative and a positive solution. Just remember (as the MGMAT guide tells you): "An even exponent hides the original sign of the base" (p.64 of the Number properties guide).

Let's take a look at your third example:
"For all integers n, the function f is defined by f(n) = a^n, where a is constant. What is the value of f(1) ?
(1) f(2) = 100
(2) f(3) = -1000."

Statement 1 tells us that a^2 = 100 --> taking the square root yields |a|= 10 --> the solution could be +10 or -10
Statement 2, though, tells that a^3 = -1000 --> an uneven exponent doesn't hide the sign --> a has to be -10

So the difference between the examples you have provided is that in the first one you don't have an expression with an even exponent while in the second and third you do.
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Re: Function and square root strategy [#permalink] New post 27 Oct 2012, 23:14
vinnik wrote:
Hi,

I have a conceptual question. It is very easy, but i am confused. This is from MGMAT, 4th edition, Equations inequalities and VIC's book.
Chapter no. 5 (functions strategy) Pg no. 73

The question is :-

If f(x) = x^3 + √x and g(x) = 4x - 3, What is f(g(3)) ?
Solution as given in the book:-
g(3) = 4(3) - 3 = 9

f(g(3)) = f(9) = 9^3 + √9 = 729 + 3 = 732

My doubt :- √9 - the square root of 9 must have two solutions, one is 3 and another is -3 (3,-3) which give two answers.
Why are we sticking to the positive solution of √9.

This strategy is used in couple of questions (Q4 and Q5) same chapter.

There is a similar question in the same chapter, pg no. 79, Q2
If g(x) = 3x + √x , What is the value of g(d^2 + 6d + 9) ?

Now here are two solutions:-
g(d^2 + 6d + 9) = 3(d^2 + 6d + 9) + √(d+3)^2
This √(d+3)^2 gives two solutions as given in the answer explanation.

+(d+3) and -(d+3) which give two answers.

One more question in O.G-12, DS, Q36.

For all integers n, the function f is defined by f(n) = a^n, where a is constant. What is the value of f(1) ?
(1) f(2) = 100
(2) f(3) = -1000.

My focus is on the (1) which clearly says that √100 has two solutions.

I really apologize if i have given a very long explanation, but i am really confused by the contradicting explanations. I hope i have clearly explained my doubt.

Looking forward to your replies.

Thanks & regards
Vinni


To my knowledge.,

If x = \sqrt{a^2}
x = a

And
If x = a
\sqrt{x} = + or - \sqrt{a}

i.e

x = \sqrt{25}
x=5

However.,

x^2 = 25
x = + or - 5

This is true not only for the GMAT but also in all general math.

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Re: Function and square root strategy [#permalink] New post 29 Oct 2012, 23:36
MacFauz wrote:

To my knowledge.,

If x = \sqrt{a^2}
x = a

And
If x = a
\sqrt{x} = + or - \sqrt{a}

i.e

x = \sqrt{25}
x=5

However.,

x^2 = 25
x = + or - 5

This is true not only for the GMAT but also in all general math.

Kudos Please... If my post helped.


Thanks guys.

I understand that i missed on this concept :-
x^2 = 25
x = + or - 5

But I am still not able to understand :-
x = \sqrt{25}
x=5

I have seen couple of examples which give different results. In one of the questions it gives one solution as explained earlier and in one of the questions it give two solutions.

I am marking this in bold. Please explain the difference between the two. Why one is giving only a single solution and other is giving two.

First Question:-
If f(x) = x^3 + √x and g(x) = 4x - 3, What is f(g(3)) ? (MGMAT -3 equations, inequalities & VIC, (functions strategy) Pg no. 73)
Solution as given in the book:-
g(3) = 4(3) - 3 = 9

f(g(3)) = f(9) = 9^3 + √9 = 729 + 3 = 732
This √9 is the biggest confusion only if i compare it with the below question.
Here, √9 = 3 and then added to 729. i.e 729 + 3 = 732.


Now, second question
If g(x) = 3x + √x , What is the value of g(d^2 + 6d + 9) ? (pg no. 79, Q2)
g(d^2 + 6d + 9) = 3(d^2 + 6d + 9) + √(d+3)^2

Now this one √(d+3)^2. How many solutions this must have ? According to me and if i compare it with the above question, it must have one solution i.e (d+3)

However, in the answer explanation there are two solutions + or - (d+3) which give two answers.
3d^2 + 19d + 30 or 3d^2 + 17d + 24


Please explain the difference between the two. I am answering questions incorrectly because of this concept.
Why there are two solutions for this \sqrt{(d+3)^2} = + or - (d+3)

and one solution for this √9 = 3 or \sqrt{3^2} = 3

Regards
Vinni
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Re: Function and square root strategy [#permalink] New post 30 Oct 2012, 06:01
vinnik wrote:

First Question:-
If f(x) = x^3 + √x and g(x) = 4x - 3, What is f(g(3)) ? (MGMAT -3 equations, inequalities & VIC, (functions strategy) Pg no. 73)
Solution as given in the book:-
g(3) = 4(3) - 3 = 9

f(g(3)) = f(9) = 9^3 + √9 = 729 + 3 = 732
This √9 is the biggest confusion only if i compare it with the below question.
Here, √9 = 3 and then added to 729. i.e 729 + 3 = 732.


Now, second question
If g(x) = 3x + √x , What is the value of g(d^2 + 6d + 9) ? (pg no. 79, Q2)
g(d^2 + 6d + 9) = 3(d^2 + 6d + 9) + √(d+3)^2

Now this one √(d+3)^2. How many solutions this must have ? According to me and if i compare it with the above question, it must have one solution i.e (d+3)

However, in the answer explanation there are two solutions + or - (d+3) which give two answers.
3d^2 + 19d + 30 or 3d^2 + 17d + 24


Please explain the difference between the two. I am answering questions incorrectly because of this concept.
Why there are two solutions for this \sqrt{(d+3)^2} = + or - (d+3)

and one solution for this √9 = 3 or \sqrt{3^2} = 3

Regards
Vinni

Now I think I know what your problem is: √9 is NOT equal to \sqrt{3^2}.

√9 = 3
This is +3 and only +3 because a square root cannot take negative values (just remember that). So there are no two solutions here.

√(d+3)^2
We're looking for the value of d; to determine it, we need to know the value of (d+3).
Look at the expression without the root: (d+3)^2 -> (d+3) could be negative or positive, depending on d.
For example, if d is -10, (d+3) = -7 and (d+3)^2 = 49.
If d is 4, then (d+3) = 7 and (d+3)^2 = 49.
So in both cases we have the same result for (d+3)^2 but two different values for d. This occurs because we square (d+3) which hides the original sign of the term. So, when you solve √(d+3)^2 you don't know which of these values is the correct one (there are two possible solutions but we do not know which one of these two is the actual solution).
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Re: Function and square root strategy [#permalink] New post 30 Oct 2012, 09:31
Alex, thanks for your reply.

Recently i revised one of the concepts from MGMAT

x^2 < 4
√x^2 < √4
|x| < 2
x < 2 or -x < 2 => x > -2
-2 < x < 2

Is this a concept that has been applied on √(d+3)^2 ?

Looking forward to your reply .

Regards
Vinni


alex01 wrote:
Now I think I know what your problem is: √9 is NOT equal to \sqrt{3^2}.

√9 = 3
This is +3 and only +3 because a square root cannot take negative values (just remember that). So there are no two solutions here.

√(d+3)^2
We're looking for the value of d; to determine it, we need to know the value of (d+3).
Look at the expression without the root: (d+3)^2 -> (d+3) could be negative or positive, depending on d.
For example, if d is -10, (d+3) = -7 and (d+3)^2 = 49.
If d is 4, then (d+3) = 7 and (d+3)^2 = 49.
So in both cases we have the same result for (d+3)^2 but two different values for d. This occurs because we square (d+3) which hides the original sign of the term. So, when you solve √(d+3)^2 you don't know which of these values is the correct one (there are two possible solutions but we do not know which one of these two is the actual solution).
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Re: Function and square root strategy [#permalink] New post 01 Nov 2012, 18:04
The way to look at it is a square of some number has two roots, i.e. 25 has two roots +\sqrt{25} and -\sqrt{25}.
But something \sqrt{25} is only positive , i.e. 5 ( with the exception of imaginary numbers that are not covered in GMAT)
Re: Function and square root strategy   [#permalink] 01 Nov 2012, 18:04
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