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
Sorry, it should be "Discontinuity."
1. Find out if there is any vertical asymptote in each respective domain, i.e. if a vertical asymptote exists for 1/(x-2) at x<1, and if one exists for 2x-3 at x≥1.
2. If there is any, then g(x) is discontinuous.
Otherwise check if it is continuous at x=1, i.e. if the limit x->1- equals the limit x->1+.
3. If 2 is satisfied, verify if g(1) exists.
If it exists, g(x) is continuous in the interval (-∞,+∞).
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
To determine the continuity of the function g(x), we need to check if it is continuous at every point within its domain.
In this case, we have two separate cases for g(x):
Case 1: x < 1
For x < 1, g(x) = 1 / (x-2). This expression represents a rational function. Rational functions are continuous everywhere in their domain except where the denominator is equal to zero. In this case, the denominator (x - 2) becomes zero when x = 2. Since x = 2 is not within the domain (x < 1), we don't have any points of discontinuity in this part of the function.
Case 2: x ≥ 1
For x ≥ 1, g(x) = 2x - 3. The function 2x - 3 is a linear function, and linear functions are continuous everywhere. Therefore, there are no points of discontinuity in this part of the function.
As we have seen, there are no points of discontinuity in either case. Therefore, the correct answer is A.) The function g(x) is continuous.