A writer uses the exposition to introduce the characters, setting, events, and any other relevant background information of a story; the exposition is always found at the beginning of a story. Expositions can be developed through narrative paragraphs, dialogues between characters, monologues, a character's disclosed thoughts, or even through "in-universe media" found in the story, such as newspaper articles, letters, or journal entries (Literary Devices, "Exposition").
Through the dialogue exchange in the first chapter of S. E. Hinton's Rumble Fish, we learn the protagonist's name is Rusty-James, called Rusty for short. On a beach, he runs into his childhood best friend Steve Hays, whom Rusty hasn't seen since Rusty was fourteen and Steve was twelve, five or six years ago. Though the story is narrated while the two are on the beach, the story itself is set in a different, urban location. Within this first chapter, we also learn that the reason why the pair have not seen each other in five or six years is because Rusty has been in a reformatory for the past five years.
In the second chapter, through narrative paragraphs and more dialogue, we begin to learn a bit more about events leading up to Rusty being sent to a reformatory. Specifically, we learn that a character named Biff Wilcox had challenged Rusty to a fight for saying something to a girl that Biff had considered to be insulting. Steve tries to dissuade Rusty from showing up at the fight and even reminds Rusty of an admonition against gang violence Motorcycle Boy once gave before Rusty cuts Steve off: "But you know what the Motorcycle Boy said about gang--" (p. 10). Motorcycle Boy is Rusty's older brother and an ex-gang leader; he is also a hero and central figure in the story. Regardless of the reminder of Motorcycle Boy's warnings, Rusty decides the best thing is to meet Biff at the appointed place backed up by his friends Steve, B.J. Jackson, and Smokey Bennet.
All in all, the exposition in these two chapters serves to introduce us to the main characters and begin to develop the central theme concerning gang violence.