Answer to Question #106691 in Calculus for Parul

Question #106691
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]
1
Expert's answer
2020-03-27T14:37:56-0400

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"


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Comments

Assignment Expert
07.09.20, 20:00

Dear Neelabh Mam, thank you for leaving a feedback.

Neelabh Mam
06.09.20, 08:31

d) The function f: R --> R , defined by f(x)= x|x| , is an odd function is true because x|x| is composed of multiplication of an odd and and even function viz x and |x|. By the basic properties of even-odd function multiplication, the product of an odd and and even function is an odd function

Assignment Expert
01.09.20, 21:17

Dear mamn. Thank you for leaving a feedback.

mamn
01.09.20, 10:58

(a) we need to give 2 examples. second one could be : { x - 2 | x ∈ Z+, x >= 9 } (b) we are given a valid function with domain RxR and an unknown range. The question is if this is a binary operation. A binary operation on R defines a function from RxR to R. Although the domain is specified to be R, the range is not specified to be R, hence this statement is generally false as in: for the binary operation a * b = sqroot(a + b) would result in imaginary numbers for say a = -1 , b = -2 etc

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