Exercise 14 Page 521

We want to find the domain, points of discontinuity, and x- and y-intercepts of the graph of the given rational function. We also want to determine whether the discontinuities are removable or non-removable. Let's do these things one at a time.

Domain

Consider the given function. y=x^2+2x/x^2+ 2 Recall that division by zero is not defined. Therefore, the rational function is undefined where x^2+ 2=0. ccc x^2+ 2=0 ⇕ x^2= -2 Notice that there is no real number that satisfies the equality x^2=-2. This means that all real numbers are included in the domain. Domain All real numbers

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 there are no points of discontinuity of our function.

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.

We found that if y=0, the value of x is -2.

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. Therefore, to find the y-intercept, we have to substitute 0 for x in the function rule and solve for y.

The y-intercept of the graph is the point (0,0).