Detarmine whether each of these functions is a bijection from R to R
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The function is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto. We also say that such a function is bijective.
a)
Let and let us assume
So,
Hence, we have implies .
So, is one-one (injective).
Also we know
So, we clearly observe the Co-Domain is the same as the Range, so is surjective.
Therefore is a bijection.
b)
We have but
So, is not one-one (injective).
Therefore is not a bijection.
c)
The function is not defined at
Therefore is not a bijection from to
d)
Let and let us assume
So,
Hence, we have implies .
So, is one-one (injective).
Also we know
So, we clearly observe the Co-Domain is the same as the Range, so is surjective.
Therefore is a bijection.