Hi Kristina. I can understand your explanations well enough, but there are some things that could be reworded.
assuming that this function is continuous
We already know function f is not continuous (because it isn't defined at x=4). You ought to say, "assuming the function is continuous on [2,3]" because that's the interval you're working with in part (a).
a root x = c on [2,3] between 2 and 3, where f(c) = 0.
It's okay to call c a root, but then it's redundant to add that f(c) is zero. You could replace the word 'root' with 'value'. That way, your statement f(c)=0 serves to identify value c as the root.
Another redundancy is to say "on [2,3] between 2 and 3" because those are both the same interval (endpoints notwithstanding). Delete the "on [2,3]" part and simply say "between x=2 and x=3".
Should I say that there "must"? There could be an undefined value between 2 and 3, for all we know right?
There are no undefined values of f(x) between x=2 and x=3 because we have already assumed the function is continuous on that interval.
And there's nothing wrong with using the word 'must'. As x goes from 2 to 3, we know that f(x) -- that is, y -- takes on
every value ranging from -5 to 10 because f is continuous there. Zero is one of those y-values. So, yes, f(x) must be 0 at some x-value within the interval [2,3].
at f(4) we have an undefined function value. We do not have enough information to prove that there is a root because the root may possibly be the point that is undefined at f(4).
Again, your intent is clear, but we can tweak the wording a bit. The symbol f(4) doesn't represent anything in this exercise, so it's better to not use it. Instead, you may say, "at x=4 the function is not defined".
Lastly, we don't want to suggest that a root could be at a point that's undefined. A better explanation would be that a root is not guaranteed because the "discontinuity might be at (4,0)".
Overall, your answers are a fine first draft. And it's a good sign that you'd recognized a need to inquire about the wordings, too.
Keep up the good work.
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