Show that the function f : R → R : f(x) = 1 + x2 is many-one into.
To prove: function is many-one into
Given: f : R → R : f(x) = 1 + x2
We have,
f(x) = 1 + x2
For, f(x1) = f(x2)
⇒ 1 + x12 = 1 + x22
⇒ x12 = x22
⇒ x12 - x22 = 0
⇒ (x1 – x2) (x1 + x2) = 0
⇒ x1 = x2 or, x1 = –x2
Clearly x1 has more than one image
∴ f(x) is many-one
f(x) = 1 + x2
Let f(x) = y such that 
⇒ y = 1 + x2
⇒ x2 = y – 1
If y = 3, as 
Then x will be undefined as we can’t place the negative value under the square root
Hence f(x) is into
Hence Proved
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