Suppose f: R^2--->R is defined by f(x,y) = 3y. Is f one-to-one? Is f onto? Is f a bijection?
Definition: A function is called one-to-one if implies where .
Definition : A function f : is called onto if for every there exit at least one element such that f(a)=b.
Definition : A function f : is called bijection if f is both one-to-one and onto.
I have used the above definitions to solve the given problem below.
is defined by f(x,y) = 3y .
Clearly but f(1,0)=f(2,0)=0.So f is not one-to-one.
Let . Now and .
So
Since b is arbitrary element in R therefore f is onto.
Since f is not one-to-one therefore f is not bijection.
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