ex111(1) a-c-eex XXXXXXXXXXa-c-e3-4-5ex33 (6)

Question

ex111(1) a-c-eex XXXXXXXXXXa-c-e3-4-5ex33 (6)

David · Accepted Answer

Answer
1. (a)  
f(x) = x5/3 
For the above function for x   y  
f(x)   f(y) 
Hence f(x) is one to one function. 
1. (c) 
Let x1 and x2 are real numbers, such that f(x1) = f(x2) , then 
f(x1) = 3x1 + 5 = f(x2) = 3x2+5 
so, we get 
x1 = x2 
Therefore f(x) is one to one and onto as well. 
f(x) and f-1(x) both are differentiable so continuous as well so it is homeomorphism.  
1(e)  
f(x) = 1/x 
Let x1 and x2 are real numbers, such that f(x1) = f(x2) , then 
f(x1) = 1/x1 = f(x2) = 1/x2 
so, we get 
x1 = x2 
Therefore f(x) is one to one and onto as well. 
f(x) and f-1(x) both are differentiable so continuous as well so it is homeomorphism.
Answer 
2. (a)  
Here since no boundaries are given for the variable x on domain R, so it is an open set. 
2. (c)  
Here in this case, 
x = 1/n where n is some natural number, hence 
So since both domain and range on the function is not known so it is neither a closed nor an open set. 
2.

ex111(1) a-c-e ex XXXXXXXXXXa-c-e 3-4-5 ex33 (6)

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