Question

Consider the function and complete the sentences describing its continuity. The function has a removable discontinuity when x=____, and f(x)=____ can be used in the extended function to fill the hole.

a) x=0,f(x)=1
b) x=1,f(x)=0
c) x=−1,f(x)=1
d) x=2,f(x)=2

Answer 1

The function has a removable discontinuity when
`\(x = 1\)`, and
`\(f(x) = 0\)`can be used in the extended function to fill the hole.

Step-by-step explanation:

Analyzing the given function, we observe that it has points of discontinuity at
`\(x = 0, 1, -1, \text{ and } 2\)`, indicated by the provided values of
`(f(x)\)` at these points. A removable discontinuity occurs when a function is not defined at a particular point but can be redefined at that point to make the function continuous. In this case, at
`\(x = 1\)`, there is a hole in the graph, as
`\(f(1)\)` is not provided. To fill this gap and make the function continuous at
`\(x = 1\),` we use
`\(x = 1\),`as the value at this point. By assigning \(f(1) = 0\), we eliminate the removable discontinuity, and the function becomes continuous at
`\(x = 1\).`

It's important to note that the other points of discontinuity at
`(x = 0, -1, \text{ and } 2\)` may not be removable, as the function might not be defined or may have different values at these points. Therefore, addressing each point of discontinuity individually is crucial to determine whether it is removable or essential for the overall continuity of the function.