Which of the following is an odd function? f(x) = x3 + 5x2 + x f(x) = x2 + x f(x) = –x

A function f(x) is said to be odd provided that f(-x) = -f(x) . 

1) f(x) = x³ + 5x² + x 

-f(x) = -(x³ + 5x² + x) = x³ - 5x² - x 

f(-x) = (-x)³ - 5(-x)² - (-x) = -x³ - 5x² + x 

f(-x) ≠ -f(x) , so this is not an odd function. 

 
2) F(x) = √x is only defined for x ≥ 0 , so it makes no sense to consider F(-x). 
This function is neither odd nor even. 

 
3) f(x) = x² + x 

-f(x) = -x² - x 

f(-x) = (-x)² + (-x) = x² - x 

f(-x) ≠ -f(x) , so this is not an odd function. 

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4) f(x) = - x 

-f(x) = x 

f(-x) = - (-x) = x 

-f(x) = f(-x) , so this is an odd function.

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