Among various properties of continuous, we have ƒ is continuous function on [a,b] and ƒ(a)ƒ(b) < 0, then there exists a point c in (a,b) such that ƒ(x) = 0 equivalently if ƒ is continuous on [a,b] and x ε R is such that ƒ(a) < x < ƒ(b) then there is c ε (a,b) such that x = f(c). It follows from the above result that the image of a closed interval under a continuous function is a closed interval.
1) The number of continuous function on R which satisfy (ƒ(x))2 = x2 for all x ε R is
(A) 1 (B) 2 (C) 4 (D) 8
2) Suppose that Æ’(12) = 1 and Æ’ is continuous on [0,1] assuming only rational value in the entire interval. The number of such functions is
(A) infinite (B) 2 (C) 4 (D) 1
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3 Answers
229I think the answers would be 1/2 and infinite
- Dwijaraj Paul Chowdhury i meant Q1- 2Upvote·0· Reply ·2013-06-17 02:55:34
Sourup Nag the answer to the first question cannot be 1/2 as it asks for the number of functions....
1357|f(x)|=|x|
The functions will be
f(x)=x
f(x)=-x
f(x)=|x|
f(x)=-|x|
Jeet Sen Sharma last 2 equations are ok.. bt d first two are nt clear...
Soumyadeep Basu For f(x)=x and f(x)=-x, (f(x))^2=x^2 as x^2=(-x)^2 for all x.