31. Show that each one of the following is an odd function:
(i) f(x) = x3 - x for all x Є R (ii) f(x) x = 1/x for all x Є R – {0}
(iii) f(x) = І x І/x for all x Є R – {0} (iv) f(x) = x3 + 3 x – 4 sin 3 x for all x Є R.

Question

31.	Show that each one of the following is an odd function:
(i)   f(x) = x3  - x for all x Є R                                  (ii) f(x) x  = 1/x for all x Є R – {0}
(iii) f(x) = І x І/x for all x Є R – {0}                         (iv) f(x) = x3 + 3 x – 4 sin 3 x for all x Є R.

anonymous · Answer

same way if f(x)=f(-x) then it is even hence not odd

zarkon · Answer

show f(-x)=-f(x)

zarkon · Answer

functions don't have to be odd or even.  They can be neither.

zarkon · Answer

so for the first one

$$f(-x)=(-x)^3-(-x)=-x^3+x=-(x^3-x)=-f(x)$$