Question

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{3}}{x^{3}-1}\)

Short answer

The graph of the function \(y=\frac{x^{3}}{x^{3}-1}\) has x and y intercepts at (0,0). It has no relative extrema and no points of inflection. The function has horizontal asymptote at y = 1 and vertical asymptote at x = 1. The domain of the function is \((-\infty, 1) \cup (1, \infty)\).

Step-by-step solution

Find the x and y-intercepts

The x-intercept can be found by setting \(y=0\) and solving for \(x\). In this case, it will be \(x=0\). The y-intercept can be found by setting \(x=0\) and solving for \(y\). In this case, \( y = 0 \).

Find the relative extrema

To find the relative extrema, we locate the derivative (\(y'\)) of the function and set it equal to zero, then solve for x. However, this function does not have relative extrema as \( y' \) never equals 0 when solving for \( x \).

Find points of inflection

Points of inflection can be found by setting the second derivative, \(y''\), equal to zero and solving for \(x\). However, when we set \( y'' = 0 \), there are no real solutions for \( x \). Hence, there are no points of inflection.

Find asymptotes

The function \( y = \frac{x^{3}}{x^{3}-1} \) has a horizontal asymptote at y = 1, since as x approaches infinity or negative infinity, the ratio of the highest degrees of the numerator and the denominator are equal (both are cubes), so the ratio is the coefficient of these, which is 1. To find the vertical asymptote, we find the values of \(x\) which make the denominator zero, so \( x = 1 \).

Determine the domain

The domain of a function is the set of all possible x-values. Since we have a rational function and the denominator cannot equal 0, we exclude that x value from the domain. So the domain is all real numbers except \(x=1\) as it makes the denominator zero. So, the domain is \((-\infty, 1) \cup (1, \infty)\).