The line g(x) has slope ...
(change in y)/(change in x) = (-18 -(-20))/(1 - 0) = 2
so can be written in slope-intercept form as
g(x) = 2x -20
The x-intercept of this line is at x=10.
0 = 2x -20 . . . . the x-intercept is where g(x) = 0
20 = 2x
10 = x
The circle also intersects the x-axis at x=10, so that will be one point that is shared by the circle and g(x). A graph shows there is also another point of intersection, (6, -8).
Yes, the linear function g(x) will intersect the circle at 2 points with positive x-coordinates.