Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = 10x$. Then (Where $\mathbb{R}$ is the set of real numbers)

To determine the properties of $f(x) = 10x$, we need to check if the function is one-one (injective) and onto (surjective).

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A function is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$.

Suppose $f(x_1) = f(x_2)$

$10x_1 = 10x_2$

$x_1 = x_2$

Therefore, $f$ is one-one.

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A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is onto if for every real number $y$, there exists an $x$ such that $f(x) = y$.

Let $y$ be any real number. Solve for $x$:

$f(x) = y$

$10x = y$

$x = \frac{y}{10}$

Since $y$ is a real number, $\frac{y}{10}$ is also a real number. Every real number can be achieved as an output.

Therefore, $f$ is onto.

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Since $f(x) = 10x$ is both one-one and onto, the function is bijective.