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Chapter 1: Problem 24

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$k(x)=1 /(x-3)$$

Short Answer

Expert verified
The graph of the function \(k(x) = 1/(x - 3)\) forms a hyperbola with a vertical asymptote at \(x = 3\). The appropriate viewing window could be \([0, 6]\) for the \(x\)-axis, and \([-10, 10]\) for the \(y\)-axis.

Step by step solution

01

Understand the function

First, notice that the function \(k(x) = 1/(x - 3)\) is a version of the simple reciprocal function \(k(x) = 1/x\), but shifted 3 units to the right. This shift is caused by subtracting 3 from x in the denominator.
02

Identify the asymptote

Next, identify the vertical asymptote. This is the value of \(x\) that makes the denominator zero, because the function is undefined at this point. For this function, the denominator is zero when \(x = 3\). Therefore, \(x = 3\) is a vertical asymptote.
03

Choose the viewing window

Since the asymptote is at \(x = 3\), select a viewing window that will clearly show this point and the behavior of the function around it. There's no fixed rule for this, but one might choose the window to be \([0, 6]\) on the \(x\)-axis, and \([-10, 10]\) on the \(y\)-axis.
04

Graph the function

Enter the function into the graphing utility and set the viewing window as chosen in the previous step. The graph should show a hyperbola with a vertical asymptote at \(x = 3\), cutting through the quadrants II and IV

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Most popular questions from this chapter

Finding a Mathematical Model In Exercises \(41-50\), find a mathematical model for the verbal statement. \(z\) varies jointly as the square of \(x\) and the cube of \(y\)

Determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept.

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