Final answer:
The rational function r(x)=(x³+8)/(x²+4) has no x-intercept because the numerator cannot be zero without the denominator also being zero. The y-intercept is found by setting x to zero, which gives the point (0, 2).
Step-by-step explanation:
To find the x-intercept and y-intercept of the rational function r(x) = (x³ + 8) / (x² + 4), we must set each variable to zero in turn and solve for the other.
Finding the x-intercept:
For the x-intercept, the value of y (which is r(x) in this case) must be zero:
0 = (x³ + 8) / (x² + 4)
This implies that x³ + 8 = 0, so x = -2. However, we must consider the domain of the rational function and note that x = -2 is not an intercept since the denominator (x² + 4) doesn't equal zero at x = -2. Thus, we realize that there is no x-intercept for this function because the numerator cannot be zero without making the denominator zero as well.
Finding the y-intercept:
To find the y-intercept, we substitute x = 0 into the function:
r(0) = (0³ + 8) / (0² + 4) = 8 / 4 = 2
The y-intercept of the function is at the point (0, 2).