Match the following lists:

Answer

`a to p; b to q; c to q,s; d to r,s`

Explanation

a. `f(x) ={(sgn x)^(sgn x)}^(n)`
`={([(1)^(1)]^(n)",",x gt 0),([(-1)^(-1)]^(n) ",",x lt 0):}={(1",",x gt 0),(-1",",x lt 0):}.`
Hence, `f(x)` is an odd function.
b. `f(x)=(x)/(e^(x)-1)+(x)/(2)+1`
`or f(-x)=(-x)/(e^(-x)-1)-(x)/(2)+1`
`=(xe^(x))/(e^(x)-1)-(x)/(2)+1`
`=(xe^(x)-x+x)/(e^(x)-1)-(x)/(2)+1`
`=x+(x)/(e^(x)-1)-(x)/(2)+1`
`=(x)/(e^(x)-1)+(x)/(2)+1`
`=f(x)`
c. `f(x)={(0",","if x is rational"),(1",","if x is irrational"):}`
`f(-x)={(0",","if -x is rational"),(1",","if -x is irrational"):}`
`={(0",","if x is rational"),(1",","if x is irrational"):}`
`=f(x)`
d. `f(x)=max [tanx, cotx}`
From the graph of function, it can be verified that `f(x)` is neither odd nor even.
Also, `f(x+pi)=max {tan(x+pi),cot(x+pi)`
`=max {tanx, cot x}`
Hence, `f(x)` is periodic with period `pi`.