The function $f: [0, \infty) \rightarrow \mathbb{R}$ defined by, $f(x) = 2x^2 + 3$, is

The function $f: [0, \infty) \rightarrow \mathbb{R}$ is defined by $f(x) = 2x^2 + 3$.

Domain: $[0, \infty)$ (all non-negative real numbers)

Codomain: $\mathbb{R}$ (all real numbers)

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To check if the function is one-one, assume $f(x_1) = f(x_2)$:

$2x_1^2 + 3 = 2x_2^2 + 3$

$2x_1^2 = 2x_2^2$

$x_1^2 = x_2^2$

Since both $x_1, x_2 \in [0, \infty)$, both are non-negative:

$x_1 = x_2$

If outputs are equal, inputs must be equal. The function is one-one.

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To check if the function is onto, find the range.

When $x = 0$:

$f(0) = 2(0)^2 + 3 = 3$

This is the minimum value.

As $x$ increases, $f(x)$ increases since $x^2$ increases.

As $x \to \infty$, $f(x) \to \infty$

Range of $f = [3, \infty)$

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Codomain: $\mathbb{R}$ (all real numbers)

Range: $[3, \infty)$ (only numbers greater than or equal to 3)

Since Range $\neq$ Codomain, the function is not onto.

Values like $f(x) = 0$, $f(x) = 1$, or $f(x) = -5$ cannot be achieved as they are less than 3.

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The function is one-one but not onto.