If f(x) = x3 - \(\frac{1}{x^3}\)then show that f(x) + f\((\frac{1}x)\

1.	If f(x) = x3 - \(\frac{1}{x^3}\)then show that f(x) + f\((\frac{1}x)\) = 0
Answer» Given f(x) = x³ - \(\frac{1}{x^3}\) Need to prove: f(x) + f\((\frac{1}{x})\) = 0 Replacing x by \(\frac{1}{x}\) we get, f\((\frac{1}{x})\) = \(\frac{1}{x^3}\) - \(\frac{1}{\frac{1}{x^3}}\) = \(\frac{1}{x^3}\) - x³ Now according to the problem, f(x) + f\((\frac{1}{x})\) = x³ - \(\frac{1}{x^3}\) + \(\frac{1}{x^3}\) - x³ ⇒f(x) + f\((\frac{1}{x})\) = 0 [proved]

If f(x) = x3 - \(\frac{1}{x^3}\)then show that f(x) + f\((\frac{1}x)\) = 0

Answer»

Given f(x) = x³ - \(\frac{1}{x^3}\)

Need to prove: f(x) + f\((\frac{1}{x})\) = 0

Replacing x by \(\frac{1}{x}\) we get,

f\((\frac{1}{x})\) = \(\frac{1}{x^3}\) - \(\frac{1}{\frac{1}{x^3}}\) = \(\frac{1}{x^3}\) - x³

Now according to the problem,

f(x) + f\((\frac{1}{x})\) = x³ - \(\frac{1}{x^3}\) + \(\frac{1}{x^3}\) - x³

⇒f(x) + f\((\frac{1}{x})\) = 0

[proved]