Which of the following function is/are periodic? `(a)f(x)={1,xi sr a t ion a l0,xi si r r a t ion a l` `(b)f(x)={x-[x];`2n

Find the value of x where f(x)={x ,ifxi sr a t ion a l1-x ,ifxi si r r a t ion a li scon t inuou sdot

If the functions f(x)a n dg(x) are defined on R rarr R such that f(x)={0, x in r a t ion a l x ,x in i r r a t ion a l a n dg(x)={O , x in i r r a t ion a l x ,x in r a t ion a l then (f-g)(x)i s (a)one-one and onto (b)neither one-one nor onto (c)one-one but not onto (d)onto but not one-one

Column I: Function, Column II: Type of function f(x)="{"sgnx")"^(sgn)}; x!=0,n is an odd integer, p. odd function f(x)=x/(e^x-1)+x/2+1 , q. even function f(x)={0,ifxi sr a t ion a l1,ifxi si r r a t ion a l , r. neither odd nor even function. f(x)=max{tanx ,cotx} , s. periodic

Which of the following statements are incorrect? 1.If f(x) and g(x) are one-one then 2. f(x)+g(x) is also one-one If f(x) and g(x) are one-one then 3. f(x)dotg(x) is also one-one If f(x) is odd then it is necessarily one-one? Ia n dI Ion l y b. I Ia n dI I Ion l y c. I I Ia n dIon l y d. I ,I Ia n dI I I

f: Rvec[-1,oo)a n df(x)=1n([|s in2x|+|cos2x|]) (where [.] is the greatest integer function.) Then, (a) f(x)h a sr a ngZ (b) f(x)i sp e r iod i c function (c) f(x)i sin v e r t i b l ein[0,pi/4] (d) f(x) is into function

Find the period (if periodic) of the following function ([.] denotes the greatest integer functions): f(x)=x-[x-b],b in R

Prove that period of function f(x)=sinx, x in R " is " 2pi.

If f(x)a n dg(x) are two positive and increasing functions, then which of the following is not always true? [f(x)]^(g(x)) is always increasing [f(x)]^(g(x)) is decreasing, when f(x) 1. If f(x)>1,t h e n[f(x)]^(g(x)) is increasing.

Consider the function f(x) satisfyig the identity f(x)+f((x-1)/x)=1+x, AA x in R-{0,1} and g(x)=2f(x)-x+1

Suppose that f is even, periodic function with period 2, and that f(x) = x for all x in the interval [0, 1]. Then draw the graph of y = f(x).