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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. \(f(0) = 0\)
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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.
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General Discussion
IanStewart
Expert
GMAT Tutor
Joined: 24 Jun 2008
Last visit: 13 Sep 2026
Posts: 4,146
Location: United States
GMAT 1: 780 Q51 V47
Expert reply
GMAT 1: 780 Q51 V47
Posts: 4,146
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.
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tonebeeze
Rather than analyzing each answer, I would like to point out how the correct answer should look like.
From the equation \(f(x) = f(x^2)\) we can get a chain of equalities between the values of the function \(f\) at different points.
So, we will be able to deduce different equalities of the type \(f(a)=f(b)\), but there is no way to find explicit values of the function in any specific point.
The correct answer should be of this form, or its equivalent \(f(a)-f(b)=0\).
Only answer B is of this type.
For any specific value of \(x\), except \(0\) and \(1\), we can start an infinite chain of equalities. For example, start with \(x=2\):
\(f(2)=f(4)=f((-2)^2)=f(-2)=f(4^2)=f(16)=f((-4)^2)=f(-4)=f(16^2)=f(256)=f((-16)^2)=f(-16)=...\).
We can see that for a given \(x\), the function \(f\) will have the same value at all the points \(x, x^2,x^4,x^8,..., -x,-x^2,-x^4,-x^8,...\)
For \(0\), we just get \(f(0)=f(0^2)\), while for \(x=1\), we have \(f(1)=f((-1)^2)=f(-1)\).
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Posts: 103
[quote="tonebeeze"]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)
Now is it always true that f(2)f(2) = f(2) OR f(2) = 1
It might be or might not be true.
B - f(16) = f(-2)
f(16) = f(4) = f(2) = f(-2)
This must be true
C. f(-2) = -f(4)
f(-2) = f(4). It might or might not be equal to negative of it.
D. f(3) = 3f(3) Again this might not be true always
E. f(0) = 0 We don't know what will be the y value of the function as expressed in option C, D and E.
Hence the answer is B.
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)
Now is it always true that f(2)f(2) = f(2) OR f(2) = 1
It might be or might not be true.
B - f(16) = f(-2)
f(16) = f(4) = f(2) = f(-2)
This must be true
C. f(-2) = -f(4)
f(-2) = f(4). It might or might not be equal to negative of it.
D. f(3) = 3f(3) Again this might not be true always
E. f(0) = 0 We don't know what will be the y value of the function as expressed in option C, D and E.
Hence the answer is B.
Posts: 1,324
We don’t know what the individual outputs are for each input value.
However, we can say that certain outputs are equal at certain inputs.
F(x) = F(x^2)
Is telling us that for any given real value of x that we input into the function, the output (call it K) will be equal to the output you get when (x)^2 is input into the function.
if x = 2
F(2) = output K
F(2^2), which is F(4) = output K as well
so we can say:
F(2) = F(4)
F(4) = F(16)
and
F(-2) = F(4)
We can chain these together using the transitive property of equations:
F(-2) = F(4) = F(2) = F(16)
and we can just look at two of them:
F(-2) = F(16)
Subtracting the F(-2) from both sides of the equation. We get:
F(16) — F(-2) = 0
Answer B
For answer E:
We can not know for sure what the precise output for the input F(0) actually is since we do not know what the function is.
We can not say for sure whether the output is actually 0 or not.
Posted from my mobile device
However, we can say that certain outputs are equal at certain inputs.
F(x) = F(x^2)
Is telling us that for any given real value of x that we input into the function, the output (call it K) will be equal to the output you get when (x)^2 is input into the function.
if x = 2
F(2) = output K
F(2^2), which is F(4) = output K as well
so we can say:
F(2) = F(4)
F(4) = F(16)
and
F(-2) = F(4)
We can chain these together using the transitive property of equations:
F(-2) = F(4) = F(2) = F(16)
and we can just look at two of them:
F(-2) = F(16)
Subtracting the F(-2) from both sides of the equation. We get:
F(16) — F(-2) = 0
Answer B
For answer E:
We can not know for sure what the precise output for the input F(0) actually is since we do not know what the function is.
We can not say for sure whether the output is actually 0 or not.
sriharimurthy
Posted from my mobile device
Bunuel can you please explain again answers other than correct answer how are they wrong still can't comprehend
Bunuel
shubhim20
It would really help if you can share what part of the given explanation you were not able to comprehend, that way your doubt can be addressed effectively.
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