In Their Eyes Were Watching God, how does the image of the black woman as "the mule of the world" become a symbol for the roles Janie chooses or refuses to play during her quest?
In responding to this question it seems to be that you need to analyse the various relationships that Janie has with her three husbands, and the kind of roles that are forced upon her or that she forms for herself as a result of her marriage to them. Although the undoubted focus of the story is on Janie's search for self, other characters, and particularly her three husbands, become an incredibly important aspect of that journey. I would argue that the image of a black woman as "the mule of the world" can only be used to describe the roles that Janie either has to adopt or are rejected by her, as clearly, by the end of the story, she returns from her experiences a woman who is finally secure in her own identity and knows who she is and the kind of roles she is willing to adopt.
Both Logan Killicks and Jody Starks thus seem to try and force conventional gender roles upon Janie. Although these two figures are very different, both only seem to stifle Janie and her desire to discover her own identity. Even the ambitions and success of Jody serves to stifle Janie and her identity, trappping her in a role of submissive wife that is symbolised by the way that Jody forces her to cover her hair. Consider the tirade that she launches at Jody as he is dying in Chapter Eight:
You wouldn't listen. You done lived wid me for twenty years and you don't half know me atall. And you could have but you was so busy worshippin' de works of yo' own hands, and cuffin' folks around in their minds till you didn't see uh whole heap uh things yuh could have.
The harshness of Janie's words to her dying husband perhaps reflects the way that her marriage to Jody resulted in her becoming "the mule of the world" that your question refers to.
However, we can definitely identify the way that in her relationship with Tea Cake, Janie actually flourishes thanks to the equality that she is given and the respect with which he pays her. Even though Janie experiences considerable hardship during her time with Tea Cake, and has to suffer the terrible loss of her third husband, even being suspected of his death, she rises above these trials and the peace which she attains as she gathers in her "shawl" at the end of the story testifies to the way that even if others treat the black woman like "the mule of the world," the black woman has the resources within her to reject and spurn such objectification and live for themselves.