Question 5 [2;2]
Investigate whether the following functions are odd or even.
(a) π(π₯) = π₯^
3
(b) π(π₯) = cos π₯
A function is even if f(βx)=f(x)f(-x)=f(x)f(βx)=f(x)
A function is odd if f(βx)=βf(x)f(-x)=-f(x)f(βx)=βf(x)
1.
f(x)=xΒ³f(x)=xΒ³f(x)=xΒ³
f(βx)=(βx)Β³=βxΒ³=βf(x)f(-x)=(-x)Β³=-xΒ³=-f(x)f(βx)=(βx)Β³=βxΒ³=βf(x)
Hence, f(x)=xΒ³f(x)=xΒ³f(x)=xΒ³ is an odd function
2.
f(x)=cosxf(x)=cosxf(x)=cosx
f(βx)=cos(βx)=cosx=f(x)f(-x)=cos(-x)=cosx=f(x)f(βx)=cos(βx)=cosx=f(x)
Hence, f(x)=cosxf(x)=cosxf(x)=cosx is an even function
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