Discuss the surjectivity of `f: R->R` given by `f(x)=x^3+2` for all `x in R`

Explanation

To discuss the surjectivity of the function \( f: \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x^3 + 2 \), we need to determine whether every element in the codomain \( \mathbb{R} \) has a corresponding element in the domain \( \mathbb{R} \) such that \( f(x) = y \) for some \( y \in \mathbb{R} \).

### Step-by-Step Solution:

1. **Understand the Function**: The function is defined as \( f(x) = x^3 + 2 \). This is a polynomial function, specifically a cubic function.

2. **Set Up the Surjectivity Condition**: For \( f \) to be surjective, for every \( y \in \mathbb{R} \), there must exist an \( x \in \mathbb{R} \) such that:
\[
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