IIT JAM Follow
February 8, 2022 12:23 am 30 pts
ed institute 19. Let S=[0, 1u[2,3) and let f:S~R be 23. Let X and Y be two non-empty sets and let defined by 2x if if xe0,1 xe|2.3) f:X >Y, 8:Y> X be two mappings.If both f and g are injective (i.e., one-to-one) If then T={f(x):xe S}. then the inverse function (a.) X and Y must be infinite sets. f:T S (b.) g = f always. (c.) One of fog:YX and g of:X-~Y is always bijective (one-to-one and onto). (a.) does NOT exist (b.) exists and is continuous (d.) There exists bijective mapping a (c.) exists and is NOT continuous h:X >Y (d.) exists and is monotonic 24. Define a function f on the real line by 20. Let f be a continuous function from [0,4] to x-Lx]- if x is not an integer,T Then 3,9.Then 0 if x is an integer (a.) There must be an x such that f(x) =4 which of the following is true: (b.) There be an such that (a.) f is periodic with period 1, ie., must 3f (x) =2x +6 fx+1) = f() for all x (c.) There be that (b.) f is continuous must an x such 2f(x) =3x+6 (c.) f is one-to-one (d.) There must be an x such that f(x) =x (d.) lim f() exists for all ae R. 21. Let f and f, be two real valued functions 25. Let f:RR be a function such that f(x+y) = f ()f(V) for all x, yeR ar f(x) =1+xg (x) where lim g (x) =1. Then the defined on the real line. Define two functions 8 and h by g(x)=max{f (). f, ()} and hx)= min{f ). f)} I0 .Then function f (x) is g)+h(x) +3g (x)h(x) (a.) e (b.) 2 f+f, ()+36 () f, ) holds for all xE R (C.) A non-constant polynomial (a.) Always (d.) Equal to 1 for all xe R. (b.) Only if f(x) = f () for allx in R 26. Let f:R >R be a continuous function satisfying fof=f. Then (c.) Only and S are both positive functions or both negative functions (a.) f must be constant (b.) f(x)=x for all x in the range of f (d.) Only if at least one of the functions f and f, is identically zero. (c.) f must be a non-constant polynomial 22. Let f: R R be a strictly increasing (d.) There is no such function 27. Let f:[0,1]-[0,1] be continuous and S(0)=0, f (1) =1 . Then, f is continuous function. If {a,} is sequence in o.1 then the sequence f(a.)} is necessarily (a.) Increasing (a.) injective, but not surjective (b.) surjective, but not injective. (b.) Bounded (c.) Convergent (c.) bijective. (d.) Not necessarily bounded (d.) surjective. please provide us the solutions of all the sums, sir
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