Question 7: Use the graph of the function g(x) to answer the following questions.
g(x) = -x<sup>2</sup> + 2x + 2
a. Is g(s) continuous in the open interval (-1, 2).
Observe the graph. We see g(x) has no gap in the open interval (-1, 2), therefore, it is continuous in the given interval.
b. Is g(x) continuous from the right at x = -1? Is limitx→-1+ = g(-1)?
To see the function being continuous or not we have,
limitx→-1+ g(x) = (-1-2+2) = -1
The limit exists at x = -1.
g(-1) = -1
limitx→-1+ g(x) = g(-1)
So it is continuous.
c. Is g(x) continuous from the left at x = 2?
Is limx→2–g(x) = g(2)
Now, g(2) = -4+4+2 = 2
limx→2–g(x) = limx→2– -4+4+2 = 2
So g(2) = limx→2–g(x)
Therefore, g(x) is continuous at x = 2 and g(x) = g(2).
d. Is g(x) continuous on the closed interval [-1, 2]?
Yes. see the fig.