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# If f(x) = (x - 1)^2 + 3, which of the following is true for all x?

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Joined: 02 Sep 2009
Posts: 47918
If f(x) = (x - 1)^2 + 3, which of the following is true for all x?  [#permalink]

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03 Aug 2018, 04:57
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Difficulty:

55% (hard)

Question Stats:

55% (02:02) correct 45% (02:02) wrong based on 53 sessions

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If $$f(x) = (x - 1)^2 + 3$$, which of the following is true for all x?

I. $$2f(x) = f(x - 1) + f(x + 1)$$

II. $$f(2 - x) = f(x)$$

III. $$f(x) = f(-x)$$

A. I only

B. II only

C. III only

D. None

E. I, II and III

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Joined: 03 Apr 2018
Posts: 7
If f(x) = (x - 1)^2 + 3, which of the following is true for all x?  [#permalink]

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Updated on: 03 Aug 2018, 05:45
B.

I started with III, since it seemed the easiest to find and that wasn't true.
$$f(-x) = (-x-1)^2 + 3 = x^2+1+2x+3 = x^2 + 2x + 4$$ NOT equal to $$f(x)$$

So E is out.

Then took II.
$$f(2-x) = (2-x-1)^2 + 3 = (-x+1)^2 + 3$$ will be equal to $$f(x)$$. Thus, B.

Is there any shorter approach?

Originally posted by greenisthecolor on 03 Aug 2018, 05:24.
Last edited by greenisthecolor on 03 Aug 2018, 05:45, edited 1 time in total.
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Re: If f(x) = (x - 1)^2 + 3, which of the following is true for all x?  [#permalink]

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03 Aug 2018, 05:28
Bunuel wrote:
If $$f(x) = (x - 1)^2 + 3$$, which of the following is true for all x?

I. $$2f(x) = f(x - 1) + f(x + 1)$$

II. $$f(2 - x) = f(x)$$

III. $$f(x) = f(-x)$$

A. I only

B. II only

C. III only

D. None

E. I, II and III

Given, $$f(x) = (x - 1)^2 + 3$$

I. $$f(x - 1) + f(x + 1)=(x-2)^2+3+x^2+3$$=$$2(x^2-2x+5)=2((x-1)^2+3))+2$$=$$2f(x)+2\neq{2f(x)}$$
II. $$f(2 - x)=(2-x-1)^2+3=(1-x)^2+3=(x-1)^2+3$$=f(x)
III. $$f(-x)=(-x-1)^2+3=x^2+2x+1+3=x^2+2x+4\neq{f(x)}$$

Ans. (B)
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Re: If f(x) = (x - 1)^2 + 3, which of the following is true for all x? &nbs [#permalink] 03 Aug 2018, 05:28
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# If f(x) = (x - 1)^2 + 3, which of the following is true for all x?

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