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Chapter 2: Problem 45

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-1}$$

Short Answer

Expert verified
The function \(g(x)=\frac{1}{x-1}\) represents a one unit horizontal shift to the right of the function \(f(x)=\frac{1}{x}\). The vertical asymptote of \(g(x)\) is now at \(x=1\), and the horizontal asymptote remains at \(y=0\).

Step by step solution

01

Understand the initial function

The function \(f(x)=\frac{1}{x}\) is a hyperbola that approaches the axes as asymptotes but never touches or crosses them. The graph opens to the right and left, with a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).
02

Understand the transformation

The given function \(g(x)=\frac{1}{x-1}\) is a transformation of the function \(f(x) = \frac{1}{x}\). In \(g(x)\), \(x\) is replaced with \(x-1\), which represents a horizontal shift or translation to the right by 1 unit. As a result of this shift, the vertical asymptote is now at \(x=1\).
03

Graph the transformed function

Draw the transformed function \(g(x)\) on the graph using the information from Step 2. The graph still opens to the right and left, similar to \(f(x)\), but it is shifted to the right by 1 unit. The vertical asymptote is now at \(x=1\), and the horizontal asymptote is still at \(y=0\).

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