is f(x)=3-x^2 a bijection?

Question

Show that it is a one to one and onto function.

Answer 1

The function is not a bijection because two different values of x
map to a single value of f(x). Each value of f(x) must map to a unique value of x and vice versa. For example, x = 1 and -1 both map to f(x)=2.
f(x)=3-x^2 is surjective (onto) because for each value of f(x) there's at least one value of x.
This can be seen in the continuity of the parabolic curve.

Answer 2

A bijection is a function that is both injective and surjective function.
To prove that it is an injective function, the rule is f(x1) = f(x2) ==> x1=x2

f(x) =3-x^2
f(x1) = f(x2)
3-x1^2 = 3-x2^2
==> x1=x2
Therefore its an injective function.

To prove its a surjective function, the rule is b = f(a) for all values of R
Eg: b = f(a)
b = 3-a^2
b+a^2 =3

Therefore its not surjective
Hence it is not bijection

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