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Match the following lists: |
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Answer» Correct Answer - `a to p; b to q; c to q,s; d to r,s` 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`. |
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