Question

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{5-3 x}{x-2}\)

Short answer

The function \(y=\frac{5-3 x}{x-2}\) has an x-intercept at \(x=\frac{5}{3}\), and a y-intercept at x = 0. It has a vertical asymptote at x = 2 and a horizontal asymptote at y = -3, with no relative extrema or inflection points. The domain is \(x ∈ R, x ≠ 2\).

Step-by-step solution

Find the intercepts

The x-intercept is found by setting y = 0 and solving for x. Setting \(\frac{5-3x}{x-2}=0\) gives x = \(\frac{5}{3}\). The y-intercept is found by setting x = 0, but in this case there is a vertical asymptote at x = 0, so the function does not intercept the y-axis.

Determine the extrema

The extrema of the function are found by taking the derivative of the function and setting it equal to zero. The derivative of \(y=\frac{5-3 x}{x-2}\) is \(-\frac{1}{(x-2)^2}\). Setting this equal to 0 does not yield any real values, so there are no relative extrema.

Find the points of inflection

The function's second derivative is \(+\frac{2}{(x-2)^3}\) where it does not equal 2 (which is a point of discontinuity). But setting this equal to zero also does not yield any real values, implying that there are no points of inflection.

Find the asymptotes

A vertical asymptote exists where the function is undefined. As \((x-2) ≠ 0\), x = 2 is a vertical asymptote. A horizontal asymptote is found by taking the limit of the function as x approaches positive and negative infinity. Thus, y = -3 is a horizontal asymptote.

Define the domain of the function

The domain of a function is the set of all potential x values. The function is undefined at x = 2, therefore the domain of the function is \(x ∈ R, x ≠ 2\).