To determine if the mapping $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x + 3$ is linear, we need to examine the properties of linearity.

A function $f(x)$ is considered linear if it satisfies the following two conditions:

1. Additivity: $f(x + y) = f(x) + f(y)$
2. Homogeneity of degree 1 (or scalar multiplication): $f(cx) = c \cdot f(x)$ for any scalar $c$.

Let's test each condition:

1. Additivity: $f(x + y) = (x + y) + 3 = x + y + 3$ On the other hand, $f(x) + f(y) = (x + 3) + (y + 3) = x + y + 6$ Since $f(x + y) \neq f(x) + f(y)$, this function does not satisfy the additivity condition.

2. Homogeneity: $f(cx) = cx + 3$ whereas $c \cdot f(x) = c \cdot (x + 3) = cx + 3c$ Since $f(cx) \neq c \cdot f(x)$, the function does not satisfy the homogeneity condition either.

Since both conditions are not met, the function $f(x) = x + 3$ is not linear. The correct answer is False.

Would you like more details or have any questions?

Here are some related questions that may deepen your understanding:

1. What properties distinguish an affine function from a linear function?
2. How would the answer change if the function was defined as $f(x) = 2x$?
3. Can a linear transformation have a constant term? Why or why not?
4. How does the graph of a linear function differ from that of an affine function?
5. What are real-world examples of linear and affine transformations?

Tip: A function with a non-zero constant term is usually affine, not linear.