Suppose
g(x) = { 1 / (x - 2) if x < 1
2x - 4 if x >/= 1
The best description concerning the continuity of g(x) is that the function
A.) is continuous
B.) has a jump discontinuity
C.) has an infinite discontinuity
D.) has a removable discontinuity
E.) None of the above
notice that as x--> 1
g(1) --> 1/(1-2) = -1
but g(1) from the second function is 2(1) - 4 = -2
so there is a "jump" from -1 to -2 in the value of g(x)
in the transition at x = 1
Does that help?
Did you sketch it?
The following page let's you graph several different functions on the same grid
http://rechneronline.de/function-graphs/
Well, well, well! Let me juggle some options for you. The function g(x) seems to have different rules for different values of x. If I had a penny for every time that happened, I'd have... well, not much, but you get the idea. Anyway, from the looks of it, when x is less than 1, g(x) has a discontinuity since 1 divided by (x - 2) is undefined. However, when x is greater than or equal to 1, the function is just a good old polynomial. So, the best description concerning the continuity of g(x) is... drumroll, please... B.) has a jump discontinuity! Ain't that a clownin' contradiction?
To determine the continuity of the function g(x), we need to check the limit of the function as x approaches 1 from the left side and the right side.
1. Limit as x approaches 1 from the left side:
For x < 1, g(x) = 1 / (x - 2). As x approaches 1 from the left side, the denominator (x - 2) approaches -1, and the value of g(x) approaches infinity (∞).
2. Limit as x approaches 1 from the right side:
For x ≥ 1, g(x) = 2x - 4. As x approaches 1 from the right side, the value of g(x) approaches -2.
Since the limits from the left and right sides do not match, the function g(x) has a jump discontinuity at x = 1.
Therefore, the correct answer is B.) g(x) has a jump discontinuity.
To determine the continuity of the function g(x), we need to check if it is continuous at every point in its domain, which is the set of all real numbers.
First, let's consider the interval x < 1. In this interval, we have g(x) = 1 / (x - 2). We can see that the function is not defined when x = 2, as it would lead to division by zero. Therefore, at x = 2, g(x) has an infinite discontinuity. This means that g(x) is not continuous in the interval x < 1.
Next, let's consider the interval x ≥ 1. In this interval, we have g(x) = 2x - 4. This is a linear function, and linear functions are continuous for all real numbers. Thus, g(x) is continuous for x ≥ 1.
Combining these intervals, we can conclude that g(x) is discontinuous at x = 2 and continuous at all other points.
Therefore, the best description concerning the continuity of g(x) is (C) has an infinite discontinuity.