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18 Nov 2009, 09:07
I have a doubt.. suppose we were to write all of them in the form of functions of x,

for eg:

1) f(x^2) = f(x)*f(x)

2) f(x) = f(x^4)

3) f(x) = -f(x^2)

then would choices 4 and five be :

4) f(x) = 3*f(x)

5) f(x) = 0

or

4) f(x) = x*f(x)

5 f(x) = x

?

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20 Nov 2009, 09:59
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Expert's post
sriharimurthy wrote:
I have a doubt.. suppose we were to write all of them in the form of functions of x, then would choices 4 and five be :

4) f(x) = 3*f(x)

5) f(x) = 0

or

4) f(x) = x*f(x)

5 f(x) = x

You cannot rewrite each answer as a function of x. If I tell you, for example, that f(2) = 2, there are infinitely many possible functions for f(x). It might be that f(x) = x, or that f(x) = 2, or that f(x) = x^2 - 2, or that f(x) = 17x - 32, or that f(x) = 3^x - 7, to give a few examples. When you are told about a specific value of a function -- for example, if you're told that f(5) = 10 -- that only gives you a single point on the graph of y = f(x) (all you know is that the graph contains the point (5,10)). It gives you no information at all about the rest of the graph, and therefore gives very little information about the definition of the function. So in the attached question, you will not be able to determine what f(x) is from any of the answer choices. Instead you need to use the property given -- that f(x) = f(x^2) -- to see which answer choice can be proven to be true. If f(x) = f(x^2), then f(-2) = f(4), and applying this again, f(4) = f(16), which makes B the correct answer. We still don't know what f(x) is, however.
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20 Nov 2009, 10:11
Thanks so much Ian.

I've waited a long time for someone to clear this doubt of mine.

Cheers.
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06 Apr 2012, 12:04
Would someone be able to explain why b is the answer?
I'm really not sure how to approach this problem...
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06 Apr 2012, 12:25
KGG88 wrote:
Would someone be able to explain why b is the answer?
I'm really not sure how to approach this problem...

If function $$f(x)$$ satisfies $$f(x) = f(x^2)$$ for all $$x$$, which of the following must be true?
A. $$f(4) = f(2)f(2)$$
B. $$f(16) - f(-2) = 0$$
C. $$f(-2) + f(4) = 0$$
D. $$f(3) = 3f(3)$$
E. $$f(0) = 0$$

We are told that some function $$f(x)$$ has following property $$f(x) = f(x^2)$$ for all values of $$x$$. Note that we don't know the actual function, just this one property of it. For example for this function $$f(3)=f(3^2)$$ --> $$f(3)=f(9)$$, similarly: $$f(9)=f(81)$$, so $$f(3)=f(9)=f(81)=...$$.

Now, the question asks: which of the following MUST be true?

A. $$f(4)=f(2)*f(2)$$: we know that $$f(2)=f(4)$$, but it's not necessary $$f(2)=f(2)*f(2)$$ to be true (it will be true if $$f(2)=1$$ or $$f(2)=0$$ but as we don't know the actual function we can not say for sure);

B. $$f(16) - f(-2) = 0$$: again $$f(-2)=f(4) =f(16)=...$$ so $$f(16)-f(-2)=f(16)-f(16)=0$$ and thus this option is always true;

C. $$f(-2) + f(4) = 0$$: $$f(-2)=f(4)$$, but it's not necessary $$f(4) + f(4)=2f(4)=0$$ to be true (it will be true only if $$f(4)=0$$, but again we don't know that for sure);

D. $$f(3)=3*f(3)$$: is $$3*f(3)-f(3)=0$$? is $$2*f(3)=0$$? is $$f(3)=0$$? As we don't know the actual function we can not say for sure;

E. $$f(0)=0$$: And again as we don't know the actual function we can not say for sure.

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06 Apr 2012, 12:48
Bunuel wrote:
We are told that some function $$f(x)$$ has following property $$f(x) = f(x^2)$$ for all values of $$x$$. Note that we don't know the actual function, just this one property of it. For example for this function $$f(3)=f(3^2)$$ --> $$f(3)=f(9)$$, similarly: $$f(9)=f(81)$$, so $$f(3)=f(9)=f(81)=...$$.

I believe it's the last part in bold that I don't get. How can F(3) = F(81)? F(3) = 9, F(9) = 81. Why is there a function before the "81"?
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06 Apr 2012, 12:52
KGG88 wrote:
Bunuel wrote:
We are told that some function $$f(x)$$ has following property $$f(x) = f(x^2)$$ for all values of $$x$$. Note that we don't know the actual function, just this one property of it. For example for this function $$f(3)=f(3^2)$$ --> $$f(3)=f(9)$$, similarly: $$f(9)=f(81)$$, so $$f(3)=f(9)=f(81)=...$$.

I believe it's the last part in bold that I don't get. How can F(3) = F(81)? F(3) = 9, F(9) = 81. Why is there a function before the "81"?

Given: $$f(x) = f(x^2)$$ --> $$f(3)=f(3^2)=f(9)$$ --> similarly $$f(9)=f(9^2)=f(81)$$.
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