Are the underlined words an infinitive or a prepositional phrase?

would you rather stay home or go to town?

a.
prepositional phrase

b.
infinitive

Read this excerpt from "the light of gandhi's lamp" by hilary kromberg inglis, in which she describes driving to the police station where her sister is being detained. to avoid looking at the blackened windows at the top of the building and thinking about the history of who had died or been tortured there, i always looked instead at the little chinese restaurant on the opposite side of the road. how does this passage affect the text? it conveys, with great subtlety, the conflict she felt over having a sister who would break the law and end up in the custody of the police. it conveys, without any implied moral judgment, that those who fought against apartheid considered themselves to be soldiers in a just war. it conveys, with an extended metaphor, the idea that being detained by the police in this society was regarded as a badge of honor. it conveys, with direct language, the view held by many anti-apartheid activists that the police often acted in brutal and unlawful ways.