Question

Which graph represents a function that has the domain (–∞, –4) ⋃ (–4, 3) ⋃ (3, ∞), has a y-intercept at (0, 3), and is increasing on the interval (0, 4)?

Answer 1

You should pick the graph that satisfies all of these at once:

- It is not defined at x = −4 and x = 3 (there must be breaks at those vertical lines — e.g. open circles or vertical asymptotes at x = −4 and x = 3), so the domain is (−∞,−4) ∪ (−4,3) ∪ (3,∞).
- It passes through (0,3) (so the point at x = 0 is on the curve and has y = 3).
- On the interval (0,4) the function is strictly increasing. Because x = 3 is not in the domain, that means the branch on (0,3) must increase up to a left‑hand value at x → 3− and the branch on (3,4) must continue increasing from a right‑hand value at x → 3+ that is larger than the left‑hand value. In other words there will be a jump upward at x = 3 (two open circles at x = 3, with the right one higher).

So choose the graph that has breaks at x = −4 and x = 3, contains the point (0,3), and shows an increasing left branch on (0,3) and an increasing right branch on (3,4) with the right branch lying above the left branch at x = 3.