Overall explanation Correct option:

Deploy the API versions as unique stages with unique endpoints and use stage variables to provide the context to identify the API versions - A stage is a named reference to a deployment, which is a snapshot of the API. You use a stage to manage and optimize a particular deployment. For example, you can configure stage settings to enable caching, customize request throttling, configure logging, define stage variables, or attach a canary release for testing.

Stage variables are name-value pairs that you can define as configuration attributes associated with a deployment stage of a REST API. They act like environment variables and can be used in your API setup and mapping templates.

With deployment stages in API Gateway, you can manage multiple release stages for each API, such as alpha, beta, and production. Using stage variables you can configure an API deployment stage to interact with different backend endpoints.

For example, your API can pass a GET request as an HTTP proxy to the backend web host (for example, http://example.com). In this case, the backend web host is configured in a stage variable so that when developers call your production endpoint, API Gateway calls example.com. When you call your beta endpoint, API Gateway uses the value configured in the stage variable for the beta stage, and calls a different web host (for example, beta.example.com). Similarly, stage variables can be used to specify a different AWS Lambda function name for each stage in your API.

Incorrect options:

Use an X-Version header to identify which version is being called and pass that header to the Lambda function - This is an incorrect option and has been added as a distractor.

Use an API Gateway Lambda authorizer to route API clients to the correct API version - A Lambda authorizer is an API Gateway feature that uses a Lambda function to control access to your API. A Lambda authorizer is useful if you want to implement a custom authorization scheme that uses a bearer token authentication strategy such as OAuth or SAML, or that uses request parameters to determine the caller's identity.

Set up an API Gateway resource policy to identify the API versions and provide context to the Lambda function - Amazon API Gateway resource policies are JSON policy documents that you attach to an API to control whether a specified principal (typically an IAM user or role) can invoke the API. You can use API Gateway resource policies to allow your API to be securely invoked requestors. They are not meant for choosing the version of APIs.