Which API allows creating an extension with write-back functionality?

The Extensions API is the correct choice for creating an extension with write-back functionality in Tableau. This API enables developers to build interactive and integrated applications that can not only retrieve data from Tableau dashboards but also allow users to input or update data directly from the dashboard. Write-back functionality is essential for scenarios where users need to alter data, such as entering new information that should reflect in a data source, facilitating a more dynamic interaction with the visualization.

This API provides the necessary capabilities to create custom visualizations and interactions, making it particularly useful for developers looking to enhance the user experience by integrating external systems or additional functionalities within Tableau. Through the Extensions API, developers can harness the power of JavaScript to connect actions in the dashboard to the underlying data, making the dashboard not just a viewing platform but also a tool for data manipulation and interaction.

In contrast, the other APIs serve different purposes: the Analytics Extensions API focuses on integrating analytics capabilities into Tableau, the Document API is used for programmatically creating or modifying workbooks and data sources, and the REST API allows for managing Tableau Server resources but does not inherently support write-back functionality within the context of dashboard extensions.