Question

Find the x- and y-intercepts of the rational function r(x) = (x³)/(8x² - 4)

Answer 1

The rational function r(x) = (x^3)/(8x^2 - 4) has an x-intercept at x = 0 and a y-intercept at y = 0. Points where the denominator equals zero are not in the domain.

Step-by-step explanation:

The x-intercepts of the rational function r(x) = \frac{x^3}{8x^2 - 4} are the points where the function crosses the x-axis, which occur when r(x) = 0. To find the x-intercepts, set the numerator of the function equal to zero and solve for x:

• \(x^3 = 0\)
• \(x = 0\)

So the x-intercept is at \(x = 0\).

The y-intercept of the function is the point where it crosses the y-axis, which occurs when x = 0. To find the y-intercept, substitute x = 0 into the function:

• \(r(0) = \frac{0}{-4}\)
• \(r(0) = 0\)

Thus, the y-intercept is at \(y = 0\).

Note that the domain of this rational function excludes x-values that make the denominator equal to zero. Since the denominator 8x^2 - 4 becomes zero when x = \pm\frac{1}{2}, these values are excluded from the domain.