Question

f(x) =loge(x^3+(1+x^6)^1/2 is odd or even function

Answer 1

Final Answer : Odd Function

Understanding:
1) When f(x) = -f(-x), then f(x) is odd function.
2) When f(x) = f(-x) , then f(x) is even function.


Steps:
1)
$f(x) = ln( {x}^{3} + \sqrt{1 + {x}^{6} } ) \\ \: \: \: \: \: \: \: \: \: \: = ln( \frac{1}{ \sqrt{1 + {x}^{6} } - {x}^{3} } ) \\ = - ln( \sqrt{1 + {x}^{6} } - {x}^{3} )$

2)
$f( - x) = ln( {( - x)}^{3} + \sqrt{1 + { (- x)}^{6} } ) \\ = ln( - {x}^{3} + \sqrt{1 + {x}^{6} } )$
3) We observe that,
$f(x) = - f( - x)$
Hence, f(x) is odd function.