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If F ( X ) = 2 X + 2 − X 2 , Then F(X + Y) F(X − Y) is Equal to (A) 1 2 [ F ( 2 X ) + F ( 2 Y ) ](B) 1 2 [ F ( 2 X ) − F ( 2 Y ) ](C) 1 4 [ F ( 2 X ) + F ( 2 Y ) ] - CBSE (Science) Class 11 - Mathematics

Question

If  $f\left( x \right) = \frac{2^x + 2^{- x}}{2}$ , then f(x + yf(x − y) is equal to

• (a) $\frac{1}{2}\left[ f\left( 2x \right) + f\left( 2y \right) \right]$

• (b)  $\frac{1}{2}\left[ f\left( 2x \right) - f\left( 2y \right) \right]$

• (c)  $\frac{1}{4}\left[ f\left( 2x \right) + f\left( 2y \right) \right]$

• (d) $\frac{1}{4}\left[ f\left( 2x \right) - f\left( 2y \right) \right]$

Solution

(a) $\frac{1}{2}\left[ f\left( 2x \right) + f\left( 2y \right) \right]$

Given: $f\left( x \right) = \frac{2^x + 2^{- x}}{2}$ Now,
f(x + yf(x − y) = $\left( \frac{2^{x + y} + 2^{- x - y}}{2} \right)\left( \frac{2^{x - y} + 2^{- x + y}}{2} \right)$

⇒ f(x + yf(x − y) = $\frac{1}{4}\left( 2^{2x} + 2^{- 2y} + 2^{2y} + 2^{- 2x} \right)$ ⇒ f(x + yf(x − y) = $\frac{1}{2}\left( \frac{2^{2x} + 2^{- 2x}}{2} + \frac{2^{2y} + 2^{- 2y}}{2} \right)$

⇒ f(x + yf(x − y) = $\frac{1}{2}\left[ f\left( 2x \right) + f\left( 2y \right) \right]$

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Solution If F ( X ) = 2 X + 2 − X 2 , Then F(X + Y) F(X − Y) is Equal to (A) 1 2 [ F ( 2 X ) + F ( 2 Y ) ](B) 1 2 [ F ( 2 X ) − F ( 2 Y ) ](C) 1 4 [ F ( 2 X ) + F ( 2 Y ) ] Concept: Functions.
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