Given three generic functions $f_{i}: mathbb{R} ightarrow mathbb{R}, i in{1,2,3}$, such that $f_{1}$ is an injection but not a surjection, $f_{2}$ is a surjection

Given three generic functions $f_{i}: \mathbb{R} \rightarrow \mathbb{R}, i \in\{1,2,3\}$, such that $f_{1}$ is an injection but not a surjection, $f_{2}$ is a surjection but not an injection, and $f_{3}$ is a bijection, consider the function

$$f: \mathbb{R} \rightarrow \mathbb{R}^{3}, \quad x \rightarrow\left(\begin{array}{c}
f_{1}(x) \\
f_{2}(x) \\
f_{3}(x)
\end{array}\right)$$
and answer the following questions, providing arguments for the answers.
- Is $f$ an injection?
- Is $f$ a surjection?
- Is $f$ a bijection?