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Show that the function f: R → R: f(x) = 1 + x2 is many-one into.

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It is given that

f(x) = 1 + x2

By substituting values we get

f(-1) = 1 + (-1)2 = 2

f(1) = 1 + 12 = 2

Hence, f is many one.

Consider y = 1 + x2

It can be written as

x = √(y – 1)

If y <1 then we know that √(y – 1) is imaginary

O ∈ R has no pre image in R

Hence, f is into.

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