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18 Nov 2009, 10:07
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
47% (01:35) correct
53%
(01:33)
wrong
based on 655
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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. \(f(0) = 0\)
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. \(f(0) = 0\)
Show SpoilerMy doubt
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
?
_________________
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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04 Jan 2011, 17:37
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If function \(f(x)\) satisfies \(f(x) = f(x^2)\) for all values of \(x\), 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 a particular function \(f(x)\) has the property \(f(x) = f(x^2)\) for all values of \(x\). Note that we don't know the specific form of the function; we only know this one property. For example, for this function, \(f(3)=f(3^2)\) or \(f(3)=f(9)\); similarly, \(f(9)=f(81)\), so \(f(3)=f(9)=f(81)=\ldots\).
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 necessarily true that \(f(2)=f(2)* f(2)\) (this will only be true if \(f(2)=1\) or \(f(2)=0\), but as we don't know the actual function, we cannot say for sure);
B. \(f(16) - f(-2) = 0\): since \(f(-2)=f(4) =f(16)=...\), we have that \(f(16)-f(-2)=f(16)-f(16)=0\), so this option is always true;
C. \(f(-2) + f(4) = 0\): \(f(-2)=f(4)\), but it's not necessarily true that \(f(4) + f(4)=2f(4)=0\) (this will only be true 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 \(f(3)=0\)? Since we don't know the actual function, we cannot say for sure;
E. \(f(0)=0\): Again, as we don't know the actual function, we cannot say for sure.
Alternatively, we can consider a function that satisfies \(f(x) = f(x^2)\) for all values of \(x\). For instance, the function can be \(f(x) = 10\), meaning that for this function, no matter what the input \(x\) is, the output will always be 10. If we interpret this as a graph, we get a horizontal line at \(y = 10\). Now, we can evaluate the options taking \(f(x) = 10\):
A. \(f(4)=f(2)*f(2)\). This option is not true, since \(10 \neq 10*10\).
B. \(f(16) - f(-2) = 0\). This option is true since \(f(16) - f(-2) = 10 - 10 = 0\).
C. \(f(-2) + f(4) = 0\). This option is not true, since \(10 + 10 \neq 0\).
D. \(f(3)=3*f(3)\). This option is not true, since \(10 \neq 3*10\).
E. \(f(0)=0\). This option is not true, since \(f(0) = 10\), not 0.
Answer: B
Hope it's clear.
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 a particular function \(f(x)\) has the property \(f(x) = f(x^2)\) for all values of \(x\). Note that we don't know the specific form of the function; we only know this one property. For example, for this function, \(f(3)=f(3^2)\) or \(f(3)=f(9)\); similarly, \(f(9)=f(81)\), so \(f(3)=f(9)=f(81)=\ldots\).
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 necessarily true that \(f(2)=f(2)* f(2)\) (this will only be true if \(f(2)=1\) or \(f(2)=0\), but as we don't know the actual function, we cannot say for sure);
B. \(f(16) - f(-2) = 0\): since \(f(-2)=f(4) =f(16)=...\), we have that \(f(16)-f(-2)=f(16)-f(16)=0\), so this option is always true;
C. \(f(-2) + f(4) = 0\): \(f(-2)=f(4)\), but it's not necessarily true that \(f(4) + f(4)=2f(4)=0\) (this will only be true 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 \(f(3)=0\)? Since we don't know the actual function, we cannot say for sure;
E. \(f(0)=0\): Again, as we don't know the actual function, we cannot say for sure.
Alternatively, we can consider a function that satisfies \(f(x) = f(x^2)\) for all values of \(x\). For instance, the function can be \(f(x) = 10\), meaning that for this function, no matter what the input \(x\) is, the output will always be 10. If we interpret this as a graph, we get a horizontal line at \(y = 10\). Now, we can evaluate the options taking \(f(x) = 10\):
A. \(f(4)=f(2)*f(2)\). This option is not true, since \(10 \neq 10*10\).
B. \(f(16) - f(-2) = 0\). This option is true since \(f(16) - f(-2) = 10 - 10 = 0\).
C. \(f(-2) + f(4) = 0\). This option is not true, since \(10 + 10 \neq 0\).
D. \(f(3)=3*f(3)\). This option is not true, since \(10 \neq 3*10\).
E. \(f(0)=0\). This option is not true, since \(f(0) = 10\), not 0.
Answer: B
Hope it's clear.
General Discussion
20 Nov 2009, 10:59
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sriharimurthy
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.
