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# If f(x) = x2, which of the following is equal to f(m + n) +

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If f(x) = x2, which of the following is equal to f(m + n) + [#permalink]  30 Jul 2018, 09:40
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Question Stats:

88% (00:30) correct 11% (00:33) wrong based on 36 sessions
If $$f(x) = x^2$$, which of the following is equal to $$f(m + n) + f(m - n)$$?

(A) $$m^2 + n^2$$
(B) $$m^2 - n^2$$
(C) $$2m^2 + 2n^2$$
(D) $$2m^2 - 2n^2$$
(E) $$m^2n^2$$
[Reveal] Spoiler: OA

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Re: If f(x) = x2, which of the following is equal to f(m + n) + [#permalink]  11 Aug 2018, 13:39
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It will save a lot of time for many problems if you keep in mind that (m+n)^2 = m^2+n^2+2mn and (m-n)^2 = m^2+n^2-2mn. Here it obviously gives you the answer:

f(m+n)+f(m-n)= m^2 + n^2 + 2mn + m^2 + n^2 - 2mn = 2m^2 + 2n^2

So C

But this also comes in handy when it comes to remembering the inequality:

m^2 + n^2 <= (m+n)^2
Retired Moderator
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Re: If f(x) = x2, which of the following is equal to f(m + n) + [#permalink]  12 Aug 2018, 06:06
Expert's post
Explanation

The problem provides the function $$f(x) = x^2$$ and asks for the quantity $$f(m + n) + f(m - n)$$.

Plug into this function twice—first, to insert m + n in place of x, and then to insert $$m - n$$ in place of x:

$$f(m + n) = (m + n)^2 = m^2 + 2mn + n^2$$

$$f(m - n) = (m - n)2 = m^2 - 2mn + n^2$$

$$(m^2 + 2mn + n^2) + (m^2 - 2mn + n^2) = 2m^2 + 2n^2$$
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Re: If f(x) = x2, which of the following is equal to f(m + n) + [#permalink]  18 Nov 2018, 16:39
Romang67 wrote:
It will save a lot of time for many problems if you keep in mind that (m+n)^2 = m^2+n^2+2mn and (m-n)^2 = m^2+n^2-2mn. Here it obviously gives you the answer:

f(m+n)+f(m-n)= m^2 + n^2 + 2mn + m^2 + n^2 - 2mn = 2m^2 + 2n^2

So C

But this also comes in handy when it comes to remembering the inequality:

m^2 + n^2 <= (m+n)^2

Is it m^2 + n^2 <= (m+n)^2
or
m^2 + n^2 < (m+n)^2 (Without = sign).
Retired Moderator
Joined: 07 Jun 2014
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GRE 1: Q167 V156
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Re: If f(x) = x2, which of the following is equal to f(m + n) + [#permalink]  19 Nov 2018, 13:26
Expert's post
AE wrote:

Is it m^2 + n^2 <= (m+n)^2
or
m^2 + n^2 < (m+n)^2 (Without = sign).

If m and n are positive or both strictly negative (i.e non 0). Then:

$$m^2 + n^2 < (m+n)^2$$ because $$(m+n)^2= m^2 + n^2 + 2mn$$ now $$2mn$$ is positive for m and n being positive.

For $$m=n=0$$ they are equal.

If m and n have different signs then the inequality flips and $$m^2+n^2 > (m+n)^2$$.
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Re: If f(x) = x2, which of the following is equal to f(m + n) +   [#permalink] 19 Nov 2018, 13:26
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