Suppose
g(x) = { 1 / (x-2) if x < 1
2x - 3 if x >/= 1
The best description concerning the continuity of g(x) is that the function
A.) is continuous
B.) has a jump discontuity
C.) has an infinite discontuity
D.) has a removable discontuity
E.) None of the above
See response to your later post.
To determine the continuity of the function g(x), we need to analyze the different parts of the function separately.
For x < 1, g(x) is defined as 1 / (x-2). However, this part of the function creates a vertical asymptote at x = 2 since the denominator becomes zero. This means that as x approaches 2 from the left side, the function will approach positive or negative infinity. Therefore, there is a vertical discontinuity or infinite discontinuity at x = 2.
For x ≥ 1, g(x) is defined as 2x - 3, which is a linear function. Linear functions are continuous over their entire domain. There are no points of discontinuity in this section.
Since there is a point of vertical discontinuity at x = 2, we can conclude that the best description concerning the continuity of g(x) is:
C.) has an infinite discontinuity.