Final answer:
To find intercepts and asymptotes of a function, we determine the x-intercepts by solving for x when the function equals zero, the y-intercept by solving for the function when x equals zero, and the vertical asymptotes by setting the denominator equal to zero. In this case, the function r(x) = 4x - 4/x² has x-intercepts, no y-intercept, and a vertical asymptote at x = 0.
Step-by-step explanation:
To find the intercepts and asymptotes for the function r(x) = 4x - 4/x², we need to determine the x-intercepts, y-intercept, and vertical asymptotes.
X-intercepts: To find the x-intercepts, we set r(x) = 0 and solve for x. In this case, we have 4x - 4/x² = 0. Multiplying through by x², we get 4x³ - 4 = 0. Solving this cubic equation gives us the x-intercepts.
Y-intercept: To find the y-intercept, we set x = 0 and solve for r(x). Plugging in x = 0, we get r(0) = 4(0) - 4/0² = -4/0, which is undefined. Therefore, there is no y-intercept.
Vertical Asymptotes: To find the vertical asymptotes, we set the denominator equal to zero and solve for x. In this case, we have x² = 0, which gives us x = 0. Therefore, the vertical asymptote is x = 0.