# Show that the fun

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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