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Suppose f: R^2--->R is defined by f(x,y) = 3y. Is f one-to-one? Is f onto? Is...

Suppose f: R^2--->R is defined by f(x,y) = 3y. Is f one-to-one? Is f onto? Is f a bijection?

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Answer #1

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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