Which mathematical theory focuses on the properties and behaviors of functions and their graphs?

The correct choice is Function Theory because it specifically studies the properties, behaviors, and relationships of functions and their graphs. This field delves into topics such as limits, continuity, differentiability, and integrability, all of which are fundamental to understanding how functions behave and how their corresponding graphs represent these behaviors visually.

Function Theory encompasses various aspects of functions, including their domains, ranges, and transformations. It also examines different types of functions, such as polynomial, trigonometric, and exponential functions, analyzing their graphs in depth to explore their characteristics and how they interact with one another.

While Linear Algebra deals with vector spaces and linear mappings between these spaces, and Real Analysis focuses more broadly on real numbers and sequences, Function Theory zeroes in specifically on functions themselves and their graphical representations, making it the most suitable answer for the question posed. Vector Theory deals with quantities that have both magnitude and direction, which does not directly focus on functions in the way Function Theory does.