A calculus problem by Ayushman Chahar

Calculus Level 3

Let f ( x ) = e x − e − x 2 f(x)=\dfrac{e^{x}-e^{-x}}{2} and f ( g ( x ) ) = x f(g(x))=x , then g ( e 1002 − 1 2 e 501 ) − 500 = ? g(\dfrac{e^{1002}-1}{2e^{501}})-500=?

Notation: e ≈ 2.718 e \approx 2.718 denotes the Euler's number.


The answer is 1.

Ayushman Chahar by Ayushman Chahar , 21, India

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1 solution

Tom Engelsman
Jun 14, 2017

In this case, f ( x ) f(x) and g ( x ) g(x) are inverse functions of each other, hence f ( g ( x ) ) = g ( f ( x ) ) = x . f(g(x)) = g(f(x)) = x. We now calculate:

g ( f ( 501 ) ) − 500 = 501 − 500 = 1 . g(f(501)) - 500 = 501 - 500 = \boxed{1}.

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