Determine whether each of these functions is a bijection from R to R (real).a) f (x) = 2x + 1b) f (x) = x2 + 1c) f (x) = x3
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Let us determine whether each of these functions is a bijection from to
a)
Let Then and thus We conclude that and hence the function is one-to-one.
For any there exists such that
and hence the function is a surjection.
Consequently, the function is a bijection.
b)
Since we conclude that this function is not one-to-one, and therefore is not a bijection.
c)
Let Then and thus We conclude that the function is one-to-one.
For any there exists such that
and hence the function is a surjection.
Consequently, the function is a bijection.