Question

Which of the following statements are true, and which are false? Give reasons for
your answers in the form of a short proof or a counterexample.
i) There are at least two ways of describing the set ...},8,7{ .
ii) Any function with domain R×R is a binary operation.
iii) The graph of every function from ]1,0[ to R is infinite.
iv) The function :f R → R , defined by = xx)x(f , is an odd function.
v) The domain of the function ,gf o where = x)x(f and −= ,x2)x(g is ]2

Answer 1

(i) there are at least two ways of describing the set {....8 ,7}  is true, for example

$${{x| x \in {N}, x \ge 7}}$$

(ii) any function with domain R X R is a binary operation is false, because a binary operation is a mathematical operation that takes two arguments and returns one result (that is, with arity two);

(iii) the graph of every function from [0,1] to R is infinite is false, since not every function is defined in the specified interval;

(iv)The function f: R --> R , defined by f(x)= x|x| , is an odd function is true

$$f(-x)=-f(x)$$ $$f(-x)=-x|-x|=-x|x|=-f(x)$$

(v)the domain of the function f ° g ,where f(x) = √x and g(x) = √ 2-x , is [- infinity,2] is true

$$f\degree g= \sqrt {\sqrt{2-x}}=\sqrt [4]{2-x}$$ $$2-x \ge0$$ $$x \leq2$$