Answer to Question #106691 in Calculus for Parul

Question

Which of the following statement are true and which are false ? Give reason for your answers in the form of a short proof or a counterexample .
a) there are at least two ways of describing the set {7,8....}
b) any function with domain R X R is a binary operation.
c) the graph of every function from [0,1] to R is infinite .
d) The function f: R --> R , defined by f(x)= x|x| , is an odd function.
e) the domain of the function f ° g ,where f(x) = √x and g(x) = √ 2-x , is [- infinity,2]

Answer 1

Solution:

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

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

b) 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);

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

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

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

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

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

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

$$2-x \ge0$$

$$x \leq2$$