The composition of two continuous function is a continuous function.

Explanation

To prove that the composition of two continuous functions is a continuous function, we will follow these steps:

### Step-by-Step Solution:

1. **Define the Functions**:
Let \( f(x) = x^2 + 2x + 1 \) and \( g(x) = x + 3 \).

2. **Check Continuity of \( f(x) \)**:
The function \( f(x) \) is a polynomial (specifically a quadratic function). Polynomials are continuous everywhere. Therefore, \( f(x) \) is continuous for all \( x \in \mathbb{R} \).

3. **Check Continuity of \( g(x) \)**:
The function \( g(x) \) is also a polynomial (a linear function). Like all polynomials, it is continuous everywhere. Thus, \( g(x) \) is continuous for all \( x \in \mathbb{R} \).

4. **Composition of Functions**:
We will consider the composition of the functions in two ways:
- First, \( f(g(x)) \)
- Second, \( g(f(x)) \)

5. **Calculate \( f(g(x)) \)**:
Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x + 3) = (x + 3)^2 + 2(x + 3) + 1 \]
Expanding this:
\[ = (x^2 + 6x + 9) + (2x + 6) + 1 \]
Combining like terms:
\[ = x^2 + 6x + 2x + 9 + 6 + 1 = x^2 + 8x + 16 \]

6. **Calculate \( g(f(x)) \)**:
Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(x^2 + 2x + 1) = (x^2 + 2x + 1) + 3 \]
Simplifying this:
\[ = x^2 + 2x + 1 + 3 = x^2 + 2x + 4 \]

7. **Conclusion**:
Both \( f(g(x)) = x^2 + 8x + 16 \) and \( g(f(x)) = x^2 + 2x + 4 \) are polynomials. Since all polynomials are continuous, both compositions are continuous for all \( x \in \mathbb{R} \).

Thus, we conclude that the composition of two continuous functions is also a continuous function.