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Math Expert V
Joined: 02 Sep 2009
Posts: 53738
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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rajatsp wrote:
Please do let me know if there are more similar questions besides the ones you mentioned.

Thank you

Rajat

Here are several more:
for-which-of-the-following-functions-does-f-x-f-2-x-155813.html
for-which-of-the-following-does-f-a-f-b-f-a-b-164979.html
for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html

Hope this helps.
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Bunuel, would you generally recommend picking numbers as a strategy for such questions (especially if the functions are for ALL X)? Or rather the algebraic approach?

Thanks!
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Math Expert V
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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bgpower wrote:
Bunuel, would you generally recommend picking numbers as a strategy for such questions (especially if the functions are for ALL X)? Or rather the algebraic approach?

Thanks!

Usually number plugging works just fine for such questions.
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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VeritasPrepKarishma wrote:
metallicafan wrote:
Is there a faster way to solve this question rather than replacing each "x" by (1-x)? Thanks!

For which of the following functions $$f$$ is $$f(x) = f(1-x)$$ for all x?

A. $$f(x) = 1-x$$
B. $$f(x) = 1-x^2$$
C. $$f(x) = x^2 - (1-x)^2$$
D. $$f(x) = (x^2)(1-x)^2$$
E. $$f(x) = x / (1-x)$$

Tip: Try to first focus on the options where terms are added/multiplied rather than subtracted/divided. They are more symmetrical and a substitution may not change the expression. I will intuitively check D first since it involves multiplication of the terms.

Agreed; did in the same way Option D returns the same term by substituting x with (1-x)

In the OA, had they expanded option D then calculation would be required

$$x^2 (1-x)^2 = [x(1-x)]^2 = (x - x^2)^2 = x^2 - 2x^3 + x^4$$

Now replacing x with (1-x) would had required serious calculation.....
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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I basically substituted and checked each option. Somehow I get this intuition that in such questions, one should actually start from option E and work backwards, rather than starting from option A and moving forward. The reason being that GMAT would not make it so easy that option A or B itself would satisfy:). Any observations? Has anyone come across such questions (on functions), where the correct answer is option A/B?

A. f(x) = 1 - x
=> f(1-x) = 1 - (1-x) = x

So, f(x) <> f(1-x)

B. f(x) = 1 - x^2
=> f(1-x) = 1 - (1-x)^2 = 1 - (1+x^2 - 2x) = 2x - x^2

So, f(x) <> f(1-x)

C. f(x) = x^2 - (1 - x)^2

=> f(1-x) = (1-x)^2 - [(1-(1-x)]^2 = (1-x)^2 - x^2

So, f(x) <> f(1-x)

D. f(x) = x^2*(1 - x)^2
=> f(1-x) = (1-x)^2 * [(1-(1-x)]^2 = (1-x)^2 * x^2

So, f(x) = f(1-x)

E. f(x) = x/(1 - x)
=> f(1-x) = (1-x)/[(1-(1-x)] = (1-x)/x

So, f(x) <> f(1-x)

Hence, D
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Big algebra is good but not prefer in these type if question. Plugin is best approach:

Assume x = 2.

$$f(2) = 2^2(1-2)^2$$
$$= 4(-1)^2 = 4*1= 4$$

$$f(1-2) or f(-1) = -1^2(1+1)^2$$
$$=1*2^2= 1*4=4$$

D is correct answer.
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Abhii46 wrote:
You need to check for every option.
Either you can replace x in every option with 1-x or you can take x = 1 and then check for which option f(1) = f(1-1) = f(0).
In A f(1) = 1-1 = 0. f(0) = 1-0 = 1.
In B f(1) = 1 - 1^2 = 0. f(0) = 1 - 0 = 1
In C f(1) = 1^2 - (1 - 1 )^2 = 1. f(0) = 0 - (1-0)^2 = 0 - 1 = -1
in D f(1) = 1(1-1)^2 = 0. f(0) = 0.(1-0)^2 = 0. ( Right answer )
And for option E you can take x = 2 because if you take x = 1 in denominator the denominator becomes zero, which makes the f(x) undefined.
f(2) = 2/(1 - 2) = -2. f(1-2) = f(-1) = -1/(1-(-1)) = -1/2.

D is the right answer.

Please give a kudo if you like my explanation.

This explanation is much more clear than the other one; now I have understood the question!!! :D

KUDOS
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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I honestly don't get this one. Please elaborate, stared at it for 2 and half hours. lol
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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g3lo18 wrote:
I honestly don't get this one. Please elaborate, stared at it for 2 and half hours. lol

Have you gone through @Bunuel's solution at for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html#p644387 ? It is a well detailed and comprehensive solution.

As for the starting point, f(x) stands for a way to mention a relation in 'x'.

When I say, y=f(x), then there is a relation between y and x (it could be a linear relation in x or quadratic relation in x or cubic relation in x etc). In standard terms, we say that y is a function of x.

Now, in a similar fashion, if I mention y=f(1-x) ---> means that y is a function of (1-x).

The question is asking for the given 5 options of f(x), which of them will follow the relation f(x)=f(1-x) ---> simply check all the given options by substituting 1-x for x.

Example, option (a): f(x)=1-x ---> substitute 1-x for x ---> f(1-x)=1-(1-x) = 1-1+x = x $$\neq$$ f(x). Keep working like this for other options and stop when you find the one that works.

Alternately, you can plug in x=0 and see which of the options gives you f(x)=f(1-x) ---> f(0) = f(1-0)=f(1).

Hope this helps.
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Given: f(x)= f(1-x)
This means if we replace x by 1 - x in the function, still the result should be same.

Checking each option:

A f(x)=1-x
f(1 - x) = 1 - (1-x) = x. Not equal to f(x)
INCORRECT

B f(x)=1-x^2
f(1-x) = 1- (1-x)^2 = 1 - (1 +x^2 - 2x). Not equal to f(x)
INCORRECT

C f(x)=x^2-(1-x)^2
f(1-x) = (1-x)^2 - (1 - 1 +x)^2 = (1-x)^2 -x^2. Not equal to f(x)
INCORRECT

D f(x)=x^2(1-x)^2
f(1-x) = (1-x)^2(1 - 1 + x)^2 = (1-x)^2*x^2.
This is equal to f(x)
CORRECT

E f(x)= x/(1-x)
f(1-x) = 1-x/1-x + x = (1-x)/x. Not equal to f(x)
INCORRECT

Correct Option: D
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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1
study wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

A. f(x) = 1 - x
B. f(x) = 1 - x^2
C. f(x) = x^2 - (1 - x)^2
D. f(x) = x^2*(1 - x)^2
E. f(x) = x/(1 - x)

Since we are not given any restrictions on the value of x, let’s let x = 1. Thus, we are determining for which of the following functions is f(1) = f(1-1), i.e., f(1) = f(0). Next, we can test each answer choice using our value x = 1.

A. f(x) = 1 - x

f(1) = 1 - 1 = 0

f(0) = 1 - 0 = 1

Since 0 does not equal 1, A is not correct.

B. f(x) = 1 - x^2

f(1) = 1 - 1^2 = 1 - 1 = 0

f(0) = 1 - 0^2 = 1 - 0 = 1

Since 0 does not equal 1, B is not correct.

C. f(x) = x^2 - (1 - x)^2

f(1) = 1^2 - (1 - 1)^2 = 1 - 0 = 1

f(0) = 0^2 - (1 - 0)^2 = 0 - 1 = -1

Since 1 does not equal -1, C is not correct.

D. f(x) = x^2*(1 - x)^2

f(1) = 1^2*(1 - 1)^2 = 1(0)= 0

f(0) = 0^2*(1 - 0)^2 = 0(2) = 0

Since 0 equals 0, D is correct.

Alternate Solution:

Let’s test each answer choice using x and 1 - x.

A. f(x) = 1 - x

f(x) = 1 - x

f(1 - x) = 1 - (1 - x) = x

Since 1 - x does not equal x, A is not correct.

B. f(x) = 1 - x^2

f(x) = 1 - x^2

f(1 - x) = 1 - (1 - x)^2 = 1 - (1 + x^2 -2x) = 2x - x^2

Since 1 - x^2 does not equal 2x - x^2, B is not correct.

C. f(x) = x^2 - (1 - x)^2

f(x) = x^2 - (1 - x)^2 = x^2 - (1 + x^2 - 2x) = 2x - 1

f(1 - x) = (1 - x)^2 - (1 - (1 - x))^2 = 1 + x^2 - 2x - x^2 = 1 - 2x

Since 2x - 1 does not equal 1 - 2x, C is not correct.

D. f(x) = x^2*(1 - x)^2

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

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

Since x^2*(1 - x)^2 equals (1 - x)^2*x^2, D is correct.

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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Most people seem to suck at explaining these function related questions. Here is a good video explaining some function skills that helps on other questions. https://www.youtube.com/watch?v=T6-Zdr5w_bE

The key to these questions is understanding that f(x) is the function. Meaning that f(3) would mean everytime you see an x you sub in a 3. Or in this case we sub in an X-3. So for instance the first question is F(x) = 1-x............. Imagine the X is a blank or a blank parentheses waiting to be filled in by a number. so F(3) = 1-3 or F(x) = 1 - X or F (blank)= 1-(blank) or F(1-x) = 1-(1-x)
Attachments Function question 6.png [ 417.34 KiB | Viewed 2656 times ]

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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Wow, Bunuel. Thank you for that explanation. After so much studying I have never seen such a simple explanation. Definitely one of those eye-opening moments. Plugging in x and x-1 to the expression must yield the same result.
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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can i choose non integers in this question ? can someone solve the D option using 1/4 as an example ?
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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pranavpal wrote:
can i choose non integers in this question ? can someone solve the D option using 1/4 as an example ?

Yes, you can plug 1/4 for x but the question is WHY work with fractions when you can use easier numbers?

D. $$f(x)=f(\frac{1}{4}) = (\frac{1}{4})^2*(1 - \frac{1}{4})^2=\frac{9}{256}$$

$$f(1-x) =f(\frac{3}{4})=(\frac{3}{4})^2*(1 - \frac{3}{4})^2=\frac{9}{256}$$
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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For which of the following functions f(x)f(x) is the relation f(f(x))=f(f(f(f(x))))f(f(x))=f(f(f(f(x)))) NOT true for at least some values of xx not equal to zero?
f(x)=−|x|f(x)=−|x|f(x)=2−xf(x)=2−xf(x)=3xf(x)=3xf(x)=4xf(x)=4xf(x)=5f(x)=5

How to arrive solution to this question
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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This question was difficult for me to understand. The following video cleared it up for me in about 30 seconds. It's actually very simple once you understand what it's asking:

Thanks everyone
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For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Think of functions in terms of inputs and outputs. We want a function for which the input of x will lead to exactly the same result as the input of (1-x). Below is a video explanation

For a function question, it is almost always best to pick numbers, and to choose numbers that are small and manageable. Lets chose 1 for x. The quantity (1-x) would therefore equal 0.

If you put each of those inputs into the function in answer choice A you'll see very quicky that the two outputs are not equal to each other. You'll see very quickly that answer B is also incorrect. Answer C takes a bit more time. The correct answer is D.

The question if easiest to answer if you create separate columns for f(x) and f(1-x). Once again, a video explanation can be found here:
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Re: For which of the following functions is f(x)=f(1-x) for all  [#permalink]

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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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Top Contributor
study wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

A. $$f(x) = 1 - x$$

B. $$f(x) = 1 - x^2$$

C. $$f(x) = x^2 - (1 - x)^2$$

D. $$f(x) = x^2*(1 - x)^2$$

E. $$f(x) = \frac{x}{(1 - x)}$$

Let's try plugging in an easy value for x. How about x = 0.
So, we can reword the question as, For which of the following functions is f(0)=f(1-0)
In other words, we're looking for a function such that f(0) = f(1)

A) f(x)=1-x
f(0)=1-0 = 1
f(1)=1-1 = 0
Since f(0) doesn't equal f(1), eliminate A

B) f(x) = 1 - x^2
f(0) = 1 - 0^2 = 1
f(1) = 1 - 1^2 = 0
Since f(0) doesn't equal f(1), eliminate B

C) f(x) = x^2 - (1-x)^2
f(0) = 0^2 - (1-0)^2 = -1
f(1) = 1^2 - (1-1)^2 = 1
Since f(0) doesn't equal f(1), eliminate C

D) f(x) = x^2(1-x)^2
f(0) = 0^2(1-0)^2 = 0
f(1) = 1^2(1-1)^2 = 0
Since f(0) equals f(1), keep D for now

E) f(x) = x/(1-x)
f(0) = 0/(1-0) = 0
f(1) = 1/(1-1) = undefined
Since f(0) doesn't equal f(1), eliminate E

Since only D satisfies the condition that f(x)=f(1-x) when x=0, the correct answer is D

Cheers,
Brent
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Test confidently with gmatprepnow.com  Re: For which of the following functions f is f(x) = f(1-x) for all x?   [#permalink] 29 Dec 2018, 09:53

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