function problem - PS

Rahul@gurome wrote:You can easily see that it is option d.
f(1-x) = (1-x)^2 * {1-(1-x)}^2 = (1-x)^2 *x^2 = x^2 *(1-x)^2 = f(x).

Rahul can you plz. explain this one. a bit further i am not able to understand it,

goyalsau wrote:

Rahul@gurome wrote:You can easily see that it is option d.
f(1-x) = (1-x)^2 * {1-(1-x)}^2 = (1-x)^2 *x^2 = x^2 *(1-x)^2 = f(x).

Rahul can you plz. explain this one. a bit further i am not able to understand it,

What the question is asking: which of the functions would have the same value if 'x' were replaced with '1-x'
Let us now consider replacing x with 1-x in each of the answer choices.

a. f(x)=1-x
Substitute x with 1-x in the function. We have f(x) = 1 - (1-x) = 1 - 1 + x = x
Clearly, f(x) = 1-x and f(1-x) = x are not the same. Eliminate this answer.

b. f(x)=1-x(squared)
Substitute x with 1-x in the function. We have f(x) = 1 - [(1-x)^2] = 1 - (1 + x^2 - 2x) = 2x - x^2
Clearly, f(x) = 1-x^2 and f(1-x) = 2x - x^2 are not the same. Eliminate this answer.

c. f(x)=x(squared) - (1-x)squared
Substitute x with 1-x in the function. We have f(x) = (1-x)^2 - [1 - (1-x)]^2 = (1 + x^2 - 2x) - x^2 = x^2 - 2x
Clearly, f(x) = x^2 - (1-x)^2 and f(1-x) = x^2 - 2x are not the same. Eliminate this answer.

d. f(x)=x(squared) (1-x) squared
Substitute x with 1-x in the function. We have f(x) = (1-x)^2 * [1 - (1-x)]^2 = (1 + x^2 - 2x) * x^2 = (1-x)^2 * x^2
If you note, (1-x)^2 * x^2 = f(x)
Clearly, f(x) = x^2 - (1-x)^2 and f(1-x) = x^2 - (1-x)^2 are the same. This is the right answer. At this point, you can mark this answer and proceed to the next question. However, for the sake of completeness, let us work out the last answer choice too.

e. f(x) = x/1-x
Substitute x with 1-x in the function. We have f(x) = (1-x) / [1 - (1-x)] = (1-x)/x
Clearly, f(x) = x/1-x and f(1-x) = (1-x)/x are not the same. In fact, one is the reciprocal of the other. Eliminate this answer.

For help, try using this site: https://www.brightstorm.com/math/algebra ... -functions

HTH

Rahul@gurome wrote:

Rahul can you plz. explain this one. a bit further i am not able to understand it.

The question is asking that is f(x) = f(1-x)?
So for all the options what you have to do is replace x on the right hand side of the equation, by (1-x). This will give you f(1-x).
Then check whether f(x) is equal to f(1-x).
Take a.
So f(1-x) = 1- (1-x) = x and f(x) = 1-x.
Since x is not equal to 1-x, option a. is ruled out.
Next take b.
f(x) = 1-x^2. So f(1-x) = 1- (1-x)^2 = 1 – (1-2x+x^2) = 2x – x^2.
Since 1- x^2 is not equal to 2x – x^2, option b. is ruled out.
Next consider c.
f(x) = x^2 – (1-x)^2 . So (1-x) = (1-x)^2 – {1-(1-x)}^2 = (1-x)^2 – x^2.
Since x^2 – (1-x)^2 is not equal to (1-x)^2 – x^2, option c. is ruled out.
Next consider d.
f(x) = x^2 *(1-x)^2. Or f(1-x) = (1-x)^2 * {1-(1-x)}^2 = (1-x)^2 * x^2.
Now since both f(x) and f(1-x) are same, option d. is the correct answer.

We don’t need to check for e. since we have already got d. as the answer.

Thanks Rahul, Functions are tough....... At least for me i can say that......

Bnow 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(squared)
c. f(x)=x(squared) - (1-x)squared
d. f(x)=x(squared) (1-x) squared
e. f(x) = x/1-x

(squared) = superscript 2

Seems like this would be a plug and play question, but I'm getting confused! Where should I review this stuff?

Since the correct answer has to work for all values of x, we can plug in our own value for x.

Plug in x=2. Then f(x) = f(2) and f(1-x) = f(1-2) = f(-1). The question becomes:

For which of the following functions does f(2) = f(-1)?

Answer choice A:
f(2) = 1-2 = -1.
f(-1) = 1-(-1) = 2.
Doesn't work.

Answer choice B:
f(2) = 1 - 2^2 = -3.
f(-1) = 1 - (-1)^2 = 0.
Doesn't work.

Answer choice C:
f(2) = 2^2 - (1-2)^2 = 4 - 1 = 3.
f(-1) = (-1)^2 - [1-(-1)]^2 = 1-4 = -3.
Doesn't work.

Answer choice D:
f(2) = 2^2 * (1-2)^2 = 4 * 1 = 4.
f(-1) = (-1)^2 * [1-(-1)]^2 = 1 * 4 = 4.
Success!

Answer choice E:
f(2) = 2/(1-2) = -2.
f(-1) = (-1)/[(1-(-1)] = -1/2.
Doesn't work.

The correct answer is D.

GMATGuruNY wrote:
Seems like this would be a plug and play question, but I'm getting confused! Where should I review this stuff?

Since the correct answer has to work for all values of x, we can plug in our own value for x.

Plug in x=2. Then f(x) = f(2) and f(1-x) = f(1-2) = f(-1). The question becomes:

For which of the following functions does f(2) = f(-1)?
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Can i say that for all function type question which ask us to determine which one is satisfying the above equation.

Can we say use the following approach .
First we will put a value of X in above equation , The equation we are given as a condition in the question.

Then we will end up with two values of x , Then we will put both the values in the given equations and determine , Which equation satisfy above equation. ( Format ) it can we any thing { equal to , Greater than , Less than , Or me be anything }

I think this is all we are doing in the above equation.
Please GURU & RAHUL, Correct me if am wrong.
Can i use this logic in all the function type problems which are like f (x) = F ( something x )