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22 Jan 2021, 08:00
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D
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Difficulty:
95%
(hard)
Question Stats:
54% (02:52) correct
46%
(03:06)
wrong
based on 127
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Let a function be defined as \(f(x) = \frac{c^x + c^{-x}}{2}\), for \(c > 0\). What is the value of \(f(x + y) + f(x - y) \)?
A. \(2f(x) + 2f(y)\)
B. \(f(x) + f(y)\)
C. \(4f(x)f(y)\)
D. \(2f(x)f(y)\)
E. \(f(x)f(y)\)
A. \(2f(x) + 2f(y)\)
B. \(f(x) + f(y)\)
C. \(4f(x)f(y)\)
D. \(2f(x)f(y)\)
E. \(f(x)f(y)\)
22 Jan 2021, 18:26
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Bunuel
Let a function be defined as \(f(x) = \frac{c^x + c^{-x}}{2}\), for \(c > 0\). What is the value of \(f(x + y) + f(x - y) \)?
A. \(2f(x) + 2f(y)\)
B. \(f(x) + f(y)\)
C. \(4f(x)f(y)\)
D. \(2f(x)f(y)\)
E. \(f(x)f(y)\)
A. \(2f(x) + 2f(y)\)
B. \(f(x) + f(y)\)
C. \(4f(x)f(y)\)
D. \(2f(x)f(y)\)
E. \(f(x)f(y)\)
\(f(x + y) + f(x - y) = \frac{c^{x+y} + c^{-x-y}}{2} + \frac{c^{x-y} + c^{y-x}}{2} = \frac{c^{x+y} + c^{-x-y} + c^{x-y} + c^{y-x}}{2}\).
We can note the answer needs the four terms summarized as \(c^{\pm x \pm y}\) and has to have the constant \(\frac{1}{2}\).
A and B both do not feature the terms \(c^{\pm x \pm y}\).
C has the four terms but the constant is 1. D has the constant \(\frac{1}{2}\) so D is the answer.
Ans: D
Updated on: 04 Jun 2025, 07:21
Originally posted by BrushMyQuant on 11 Dec 2021, 08:33.
Last edited by BrushMyQuant on 04 Jun 2025, 07:21, edited 3 times in total.
Last edited by BrushMyQuant on 04 Jun 2025, 07:21, edited 3 times in total.
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\(f( x + y ) = \frac{c^{x + y} + c^{-(x + y)}}{2} + \frac{c^{x - y} + c^{-(x - y)}}{2}\)
= \( \frac{c^{x} * c^{y} + c^{-x} * c^{-y} + c^{x} * c^{-y} + c^{-x} * c^{y} }{2}\)
= \( \frac{c^{x} * c^{y} + c^{x} * c^{-y} + c^{-x} * c^{y} + c^{-x} * c^{-y} }{2}\)
= \( \frac{c^{x} * (c^{y} + c^{-y}) + c^{-x} * (c^{y} + c^{-y})}{2}\)
= \( \frac{ (c^{x} + c^{-x}) * (c^{y} + c^{-y})}{2}\)
= \( 2 * \frac{ (c^{x} + c^{-x}) }{2} * \frac{ (c^{y} + c^{-y})}{2}\)
= 2 * f(x) * f(y)
So, Answer will be D
Hope it helps!
Watch the following video to MASTER Functions and Custom Characters
= \( \frac{c^{x} * c^{y} + c^{-x} * c^{-y} + c^{x} * c^{-y} + c^{-x} * c^{y} }{2}\)
= \( \frac{c^{x} * c^{y} + c^{x} * c^{-y} + c^{-x} * c^{y} + c^{-x} * c^{-y} }{2}\)
= \( \frac{c^{x} * (c^{y} + c^{-y}) + c^{-x} * (c^{y} + c^{-y})}{2}\)
= \( \frac{ (c^{x} + c^{-x}) * (c^{y} + c^{-y})}{2}\)
= \( 2 * \frac{ (c^{x} + c^{-x}) }{2} * \frac{ (c^{y} + c^{-y})}{2}\)
= 2 * f(x) * f(y)
So, Answer will be D
Hope it helps!
Watch the following video to MASTER Functions and Custom Characters
