We have fully factored the denominator.
y=2x^2+5/x(x- 2)
Recall that division by zero is not defined. Therefore, the rational function is undefined where x- 2=0 and where x=0.
ccc
x- 2=0 & and & x=0
⇕ & & ⇕
x= 2 & and & x= 0
This means that neither x= 2 nor x= 0 are included in the domain.
Domain
All real numbers except x= 2 and x= 0
Points of Discontinuity
If a real number a is not in the domain of a function, then the function has a point of discontinuity at x=a. Since the domain is all real numbers except x= 2, and x= 0, we can write the points of discontinuity of our function.
Points of Discontinuity
x= 2 and x= 0
The numerator is already factored, but we cannot cancel out common factors between the numerator and the denominator. Therefore, every point of discontinuity is a non-removable discontinuity.
Intercepts
The intercepts of a function are the points where the graph intersects the axes. Let's calculate the intercepts of the given function, if there are any.
x-intercepts
The x-intercepts of a function are the points where the graph intersects the x-axis. At these points, the value of the y-coordinate is zero. Therefore, to find the x-intercepts, we have to substitute 0 for y in the function rule and solve for x.
Notice that x^2 cannot have negative value. Therefore, there are no x-intercepts.
y-intercept
The y-intercept of a function is the point where the graph intersects the y-axis. At this point, the value of the x-coordinate is zero. Since x=0 is not in the domain of the function, there is no y-intercept.
