ALPHA DIVISION

Question 97

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Question

The following four functions have various symmetries. Let S have an initial value of 0. Add 1 to S for each function symmetric with respect to the x-axis. Add 2 to S for each function symmetric with respect to the y-axis. Add 3 to S for each function symmetric with respect to the origin. Compute S.

I.\quad a(x)=x\sin x

II.\quad b(x)=|x|\cos x+x^3

III.\quad c(x)=0

x\geq 0

;

c(x)=1

x<0

IV.\quad d(x)=[x]+c(x)

, where c(x) is defined above and [x] denotes the greatest integer

\leq x

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I.\quad a(x)=x\sin x

II.\quad b(x)=|x|\cos x+x^3

III.\quad c(x)=0

x\geq 0

;

c(x)=1

x<0

IV.\quad d(x)=[x]+c(x)

, where c(x) is defined above and [x] denotes the greatest integer

\leq x