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

We are given g(x) = 1/x on the interval [-1, 1], with g(-1) = -1 and g(1) = 1, and asked which statement is correct.
Option 1: 'By the Squeeze Theorem we know there exists a number c in (-1, 1) such that g(c) = 0.' This is false because 1/x never equals zero for any x ≠ 0, so the equation g(c) = 0 has no solution. Moreover, the Squeeze Theorem is not the tool here since we do not have a pair of bounding functions that force g to be zero at some point.
Option 2: 'By the Intermediate Value Theorem we know there exists a number c in (-1, 1) such that g(c) = 0.' This is also false. The IVT requires the function to be continuous on [a, b] and to take values of opposite sign at the endpoints to guarantee a root. However, g is not even defined at x = 0, so it is not continuous on the entire interval [-1, 1], so IVT cannot be applied.
Option 3: 'The Squeeze Theorem does not apply to this function on this interval.' This statement is true. The Squeeze Theorem requires a function to be bounded between two other functions that converge to the same limit, typically with a well-behaved (continuous) context. Since g(x) = 1/x is unbounded near x = 0 and not defined there, there is no natural Squeeze setup on [-1, 1] that would yield a conclusion, so the Squeeze Theorem does not apply here.
Option 4: 'By the Intermediate Value Theorem we know that g(0) = 0.' This is false because g(0) is undefined (the function has a vertical asymptote at x = 0), so statements about the value at 0 are not meaningful within the function as given.
Option 5: 'By the Squeeze Theorem we know lim_{x->0} g(x) = 0.' This is false because lim_{x->0} (1/x) does not exist: from the left it tends to -∞ and from the right to +∞, so no finite limit is obtained. The Squeeze Theorem cannot rescue this since there is no pair of functions squeezing 1/x to 0 around x = 0.
Option 6: 'The Intermediate Value Theorem does not apply to this function on this interval.' This is true as well, but note that it is a different way of stating the same practical issue as in Option 2: IVT requires continuity on the interval, which is violated by the discontinuity at x = 0. Since the question asks which statement is correct, both Option 3 and Option 6 correctly claim that IVT or related theorems do not apply due to discontinuity, but the statement in Option 3 specifically addresses the applicability of the Squeeze Theorem, which is also correct for this function on the given interval.
In summary, evaluating each option shows that the claims relying on the IVT are invalid due to the discontinuity at x = 0, and the Squeeze Theorem does not have the proper framework to apply here, making Option 3 the correct choice among the given statements.

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?单项选择题

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