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Chapter 9: Problem 41

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

Short Answer

Expert verified
The graph of the function \(y=(x-3)/x\) has an x-intercept at (3,0), no y-intercept, a horizontal asymptote at \(y=1\) and a vertical asymptote at \(x=0\). The function has no extrema and no inflection points. The domain is \((-鈭�, 0) 鈭� (0, +鈭�)\).

Step by step solution

01

Find the Intercepts

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). However, we notice that substituting \(x = 0\) is undefined. Hence, the function does not have a y-intercept. To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). We find \(x\) equals to 0.
02

Identify the Asymptotes

Vertical asymptotes occur where the function is undefined. From the equation, it's clear that the function is undefined at \(x = 0\), so we have a vertical asymptote at \(x = 0\). To find the horizontal asymptote, we look at the degree of the numerator and the denominator. Since they are the same, we divide the coefficients of the highest degree terms, which results in \(y = 1\). Thus, the horizontal asymptote is at \(y = 1\).
03

Find the Extrema and Inflection points

The extrema and inflection points are found by taking the first and second derivatives. However, the simplified derivative of the function remains positive for all \(x\) excluding 0. Therefore, there are no extrema in the function. The second derivative does not change sign, indicating that the graph of the function has no inflection points.
04

State the Domain

The domain of the function consists of all possible \(x\)-values. This function is defined for every real number except \(x = 0\). Therefore, the domain is \((-鈭�, 0) 鈭� (0, +鈭�)\).
05

Sketch the Graph

Now, plot the obtained points, vertical and horizontal asymptote. Because the function is always increasing, the graph lies below the horizontal asymptote to the left of the vertical asymptote and above the horizontal asymptote to the right.

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Most popular questions from this chapter

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{3}+3 x^{2}+9 x-2\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4 / 3}\)

The cost \(C\) (in dollars) of removing \(p \%\) of the air pollutants in the stack emission of a utility company that burns coal is modeled by \(C=80,000 p /(100-p), \quad 0 \leq p<100\) (a) Find the costs of removing \(15 \%, 50 \%\), and \(90 \%\). (b) Find the limit of \(C\) as \(p \rightarrow 100^{-}\). Interpret the limit in the context of the problem. Use a graphing utility to verify your result.

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=(1-x)^{2 / 3}\)

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{4}}{x^{4}-1}\)
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