A vector space over $\mathbb{R}$ is a set $V$ of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following $10$ axioms. The axioms must hold for all vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ in $V$ and for all scalars $\alpha, \beta$ in $\mathbb{R}$.

1. (Closed under addition:) The sum of $\mathbf{u}$ and $\mathbf{v}$, denoted $\mathbf{u} + \mathbf{v}$, is in $V$.

2. (Closed under scalar multiplication:) The scalar multiple of $\mathbf{u}$ by $\alpha$, denoted $\alpha \mathbf{u}$, is in $V$.

3. (Addition is commutative:) $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.

4. (Addition is associative:) $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$.

5. (A zero vector exists:) There exists a vector $\mathbf{0}$ in $V$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$.

6. (Additive inverses exist:) For each $\mathbf{u}$ in $V$, there exists a $\mathbf{v}$ in $V$ such that $\mathbf{u} + \mathbf{v} = \mathbf{0}$. (We write $\mathbf{v} = -\mathbf{u}$.)

7. (Scaling by $1$ is the identity:) $1 \mathbf{u} = \mathbf{u}$.

8. (Scalar multiplication is associative): $\alpha (\beta \mathbf{u}) = (\alpha \beta) \mathbf{u}$.

9. (Scalar multiplication distributes over vector addition:) $\alpha (\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v}$.

10. (Scalar addition is distributive:) $(\alpha + \beta) \mathbf{u} = \alpha \mathbf{u} + \beta \mathbf{u}$.

Let $V$ be the set of functions $f : \mathbb{R} \to \mathbb{R}$. For any two functions $f, g$ in $V$, define the sum $f+g$ to be the function given by $(f+g)(x) = f(x)+g(x)$ for all real numbers $x$. For any real number $c$ and any function $f$ in $V$, define scalar multiplication $c f$ by $(cf)(x) = c f(x)$ for all real numbers $x$.

Answer the following questions as partial verification that $V$ is a vector space.

(Addition is commutative:) Let $f$ and $g$ be any vectors in $V$. Then $f(x) + g(x) =$ for all real numbers $x$ since adding the real numbers $f(x)$ and $g(x)$ is a commutative operation.

(A zero vector exists:) The zero vector in $V$ is the function $f$ given by $f(x) =$ for all $x$.

(Additive inverses exist:) The additive inverse of the function $f$ in $V$ is a function $g$ that satisfies $f(x) + g(x) = 0$ for all real numbers $x$. The additive inverse of $f$ is the function $g(x) =$ for all $x$.

(Scalar multiplication distributes over vector addition:) If $c$ is any real number and $f$ and $g$ are two vectors in $V$, then $c(f+g)(x) = c( f(x) + g(x) ) =$ 