Consider the function g(x)=
1

x
on the interval [−1,1]. We know that g(−1)=−1 and g(1)=1. Which of the following statements is correct?

Question

Consider the function g(x)=
1

x
  on the interval [−1,1]. We know that g(−1)=−1 and g(1)=1. Which of the following statements is correct?

BlackTom AI · Accepted Answer

To begin, let's clarify the problem setup and what the Intermediate Value Theorem (IVT) requires. The function g is described as g(x) = 1/x on the interval [-1, 1]. A key detail is that 1/x is not defined at x = 0, so g is not defined (and hence not continuous) on the entire closed interval [-1, 1].

Next, consider what IVT needs: a function must be continuous on [a, b] in order for IVT to guarantee that it takes every value between g(a) and g(b) somewhere on [a, b]. If the function is not continuous anywhere in the interval, or not defined at a point within the interval, IVT cannot be applied.

Now, evaluate the proposed statement: "The Intermediate Value Theorem does not apply to this function on this interval." Since g(x) = 1/x fails to be defined at x = 0, it is not continuous on the entire interval [-1, 1]. This breaks the hypothesis of IVT, so the theorem cannot be applied in this case. In short, the lack of definition at 0 directly undermines the applicability of IVT here.

Exploring potential misconceptions:
- One might think IVT can still be used if g(-1) = -1 and g(1) = 1. However, those endpoint values alone do not satisfy IVT because the required condition is continuity on the entire interval, not just matching endpoint signs or values.
- Another incorrect idea is that IVT could apply if the function is continuous on subintervals that exclude the problematic point. IVT, as stated, requires continuity on the whole closed interval [a, b], so a single discontinuity within the interval disqualifies its direct application to the entire interval.

In light of these points, the given statement accurately reflects the limitation: IVT does not apply to g on [-1, 1] due to the function’s undefined/discontinuous point at x = 0.

MATH-112-301-001 Unproctored Midcourse 1 Practice Exam 1

Consider the function g(x)= 1 x on the interval [−1,1]. We know that g(−1)=−1 and g(1)=1. Which of the following statements is correct?

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