Question

Rhianna says she can draw different functions that have the same x‑intercepts and the same domain and range. Her teammates say, “No, that’s impossible!” But Rhianna insists, “It is possible, we just need to sketch the graphs.” Homework Help ✎
What if the x‑intercepts are (–5, 0), (2, 0), and (6, 0), the domain is
–5 ≤ x ≤ 7, and the range is –4 ≤ y ≤ 10? Is there more than one possible function? Give examples to help explain why or why not.
What if the x‑intercepts are (–4, 0) and (2, 0), the domain is all real numbers, and the range is y ≥ –8? Is there more than one possible function? Give examples of multiple functions or explain why there can be only one.

Answer 1

In both cases there are more than one possible function sutisfying given data.

1. If

• x‑intercepts are (–5, 0), (2, 0), and (6, 0);
• the domain is –5 ≤ x ≤ 7;
• the range is –4 ≤ y ≤ 10,

then (see attached diagram for details) you can build infinetely many functions. From the diagram you can see two graphs: first - blue graph, second - red graph. Translating their maximum and minimum left and right you can obtain another function that satisfies the conditions above.

2. If

• x‑intercepts are (–4, 0) and (2, 0);
• the domain is all real numbers;
• the range is y ≥ –8,

then you can also build infinetely many functions. From the diagram you can see two graphs: first - blue graph, second - red graph. Translating their  minimum left and right you can obtain another function that satisfies the conditions above.

Note, that these examples are not unique, you can draw a lot of different graphs of the functions.

Answer: yes, there are more than one possible function