Place brisket in a Dutch oven. Mix ketchup with enough water to make 1-1/2 cups of sauce; pour over brisket. Scatter onions around meat. Sprinkle brisket with garlic powder and paprika. Cover; simmer over medium-low heat for 2 hours, or until fork-tender. Let stand 15 minutes before carving. Serve with sauce.
Soak apricots in 1 cup cold water for 2 hours. Simmer apricots in their soaking liquid for 10 minutes, or until tender; stir in brown sugar.
Meanwhile, in a steamer over boiling water, heat sweet potatoes, covered, for 15 minutes, or just until tender. Let cool, uncovered.
Heat oven to 350°F. In a buttered 2-quart baking dish, alternate layers of potato slices and apricots in sauce; cover. Bake until heated through, basting occasionally.