The image you've uploaded contains a mathematical problem involving integrals and areas. The problem describes a function $y = F(x)$ which is the antiderivative of $y = f(x)$. The graph shows two regions labeled $A$ and $B$ under the curve $y = f(x)$, and the areas of these regions are given by:

$S_A = 5, \quad S_B = 2.$

Additionally, it mentions that $F(-2) = 1$, and the task is to find $F(1)$.

To solve this, we can use the relationship between a function and its antiderivative. Since $F(x)$ is the antiderivative of $f(x)$, we know:

$F(b) - F(a) = \int_a^b f(x) \, dx$

Let’s break down the solution step by step:

1. The area of region $A$ corresponds to the integral from $x = -2$ to $x = 0$, so:

$S_A = \int_{-2}^0 f(x) \, dx = 5$

2. The area of region $B$ corresponds to the integral from $x = 0$ to $x = 1$, so:

$S_B = \int_0^1 f(x) \, dx = 2$

3. From the properties of the antiderivative $F(x)$, we can compute $F(1)$ as:

$F(1) - F(-2) = \int_{-2}^1 f(x) \, dx$

This integral is the sum of the areas of regions $A$ and $B$:

$F(1) - 1 = S_A + S_B = 5 + 2 = 7$

So:

$F(1) = 1 + 7 = 8$

Thus, $F(1) = 8$.

Let me know if you'd like more details or further steps!

Here are some follow-up questions to deepen the understanding:

1. How can the properties of definite integrals help in solving such problems?
2. What is the difference between a function and its antiderivative in terms of graphical interpretation?
3. How would the solution change if the area of region A were different, say $S_A = 7$?
4. How does the Fundamental Theorem of Calculus apply here?
5. Can the value of $F(x)$ be determined at other points given the same information?

Tip: When calculating areas using definite integrals, always break the region into smaller parts for clarity and easier computation.