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Use the definition of Odd and Even functions to determine whether each of the following functions is Odd, Even, or neither. Must show your work for credit II! 1) \( f(x)=\frac{x}{1-x^{3}} \) 2) \( f(x)=\frac{x^{2}}{1+x} \) 3) \( \quad f(x)=x-|x| \)

Monjur Alahi

Use the definition of Odd and Even functions to determine whether each of the following functions is Odd, Even, or neither. Must show your work for credit II! 1) \( f(x)=\frac{x}{1-x^{3}} \) 2) \( f(x)=\frac{x^{2}}{1+x} \) 3) \( \quad f(x)=x-|x| \)


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Salim Ahmed Kawsar

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(1) f(x) = \(\frac{x}{1-x^3}\) 

f(-x) = \(\frac{-x}{1-(-x)^3}\) 

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

∴ f(x) is neither even nor odd function.

(2) f(x) = \(\frac{x^2}{1+x}\) 

∴ f(-x) = \(\frac{(-x^2)}{1+(-x)}\) 

= \(\frac{x^2}{1-x}\) ≠ -f(x) or f(x).

(3) f(x) = x-|x|

∴ f(-x) = -x-|-x|

= -x-|x|

= -(x+|x|) ≠ -f(x) or f(x)

∴ f(x) is neither even nor odd function.

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