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# If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the

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If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the  [#permalink]

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16 Sep 2019, 21:47
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38% (01:39) correct 62% (02:51) wrong based on 12 sessions

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If function f(x) satisfies $$f(1 - x) = f(2 + x)$$ for all x, which of the following can be true?

A. $$f(x) = 3-x$$

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

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

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

E. $$f(x) = \frac{1-x}{2 + x}$$

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Re: If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the  [#permalink]

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16 Sep 2019, 22:35
A. f(x)=3−xf(x)=3−x ....

B. f(x)=1−x2f(x)=1−x2

C. f(x)=2−x−x2f(x)=2−x−x2

D. f(x)=2x−(1−x)2f(x)=2x−(1−x)2.....after substitution.....OA

E. f(x)=1−x2+x
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Re: If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the  [#permalink]

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Updated on: 17 Sep 2019, 03:00
f(1-x)=f(2+x) means that
x=0 --> f(1)=f(2)
x=1 --> f(-1)=f(),
x=2 --> f(-1)=f(4), etc
It is clear that f(x) has to be symmetrical at x=1.5.

A. f(x)=3−x
NOT symmetrical at x=1.5

B. f(x)=1−x^2
NOT symmetrical at x=1.5. In fact, f(x) is symmetrical at x=0.

C. f(x)=2−x−x^2=(2+x)(1-x)
NOT symmetrical at x=1.5. In fact, f(x) is symmetrical at x=-0.5.

D. f(x)=2x−(1−x)^2= 2x-(x^2-2x+1)=-(x^2+1)
NOT symmetrical at x=1.5. In fact, f(x) is symmetrical at x=0.

E. f(x)=(1−x)^2/(2+x)
Symmetrical at x=1.5

Originally posted by chondro48 on 16 Sep 2019, 22:39.
Last edited by chondro48 on 17 Sep 2019, 03:00, edited 4 times in total.
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Re: If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the  [#permalink]

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16 Sep 2019, 23:26
If function f(x) satisfies $$f(1 − x) = f(2 + x)$$ for all x, which of the following can be true?

Looking at the options we can eliminate B, C and D.

Checking for option A gives $$f(1 - x) = 2 + x$$ and $$f(2 + x) = 1 – x$$.

Option E gives $$f(1 − x) ≠ f(2 + x)$$

A. $$f(x) = 3 − x$$

B. $$f(x) = 1 − x^2$$

C. $$f(x) = 2 – x − x^2$$

D. $$f(x) = 2x − (1 − x)^2$$

E. $$f(x) = \frac{(1 − x)}{(2 + x)}$$

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Re: If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the  [#permalink]

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17 Sep 2019, 03:23
this problem can be solved by simple number picking: plug in numbers.

Let's take x=2

1) f(x)=3−x
-> 3-2=1

Now apply x=1 in the question stem
f(1−x)=f(2+x)
-> 1-1 =0 and 2=1+3 These two are obviously not equal so A is not the right option

B) f(x)=1−(x)^2
By plugin,
1- (2)^2 = -3
Now putting this in the question stem
f(1−x)=f(2+x)
1-(-3)=4 and 2+ (-3) = -1

Continuing this for every option we arrive on the right answer -
Re: If function f(x) satisfies f(1 - x) = f(2 + x) for all x, which of the   [#permalink] 17 Sep 2019, 03:23
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