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

Answer:

The function is not injective.

The function is not surjective.

The function is not bijective.

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:

$f(x) = y$

$y = x^{2} + 3$

$x^{2} = y - 3$

$x = \sqrt{y-3}$

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.