Question

How to determine if the function f(x) = x^2 + 3 from real numbers to real numbers is Injective, surjective, or bijective

Answer 1

The function is not injective.

The function is not surjective.

The function is not bijective.

Step-by-step explanation:

A function f(x) is injective if, and only if,
`a = b` when
`f(a) = f(b)`.

So

`f(x) = x^(2) + 3`

`f(a) = f(b)`

`a^(2) + 3 = b^(2) + 3`

`a^(2) = b^(2)`

`a = \pm b`

Since we may have
`f(a) = f(b)` when, for example,
`a = -b`, the function is not injective.

A function f(x) is surjective, if, and only if, for each value of y, there is a value of x such that
`f(x) = y`.

We have that y is composed of all the real numbers.

Here we have:

`f(x) = y`

`y = x^(2) + 3`

`x^(2) = y - 3`

`x = √(y-3)`

There is only a value of x such that
`f(x) = y` for
`y \geq 3`. So the function is not surjective.

A function f(x) is bijective when it is both injective and surjective. So this function is not bijective.