Question

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.
(1) There are at least two ways of describing the set {7, 8....}.
(2) Any function with domain R®R is a binary operation.
(3) The graph of every function from [0, 1] to R is infinite.
(4) The function f:R®R, defined by f(x) =x|x|, is an odd function.
(5) The domain of the function f(g(x)), where f(x) =√x and g(x) =√2-x, is [-infinite,2.

Answer 1

1. True. (a) Integers greater than or equal to 7, (b) $\{a_n|a_n = a_{n-1}+1\}, a_0 = 7, n = 1,2,...\\$
2. False. By definition, a binary operation on a set S is a mapping of the elements of the Cartesian product S × S to S. Take R as S and see that the statement is not always true because not all operations are defined on R completely.
3. False. A counter example: $y = x^2$ is finite in any point of [0,1].
4. True. Consider $f(-x) = (-x)|-x| = -x|x| = -f(x)$. Thus, by definition, f(x) is an odd function.
5. True. The domain of g(x) is (-infinity,2] and the domain of f(x) is x>0, which holds automatically for any x, as far as g(x) is always greater than zero. Thus, the domain of the result is (-infinity,2] .