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Chapter 2: Problem 47

(a) state the domains of \(f\) and \(g,\) (b) use a graphing utility to graph \(f\) and \(g\) in the same viewing window, and (c) explain why the graphing utility may not show the difference in the domains of \(f\) and \(g .\) $$f(x)=\frac{x-2}{x^{2}-2 x}, \quad g(x)=\frac{1}{x}$$

Short Answer

Expert verified
The domain of \(f(x)\) is all real numbers except 0 and 2, whereas the domain of \(g(x)\) is all real numbers except 0. A graphing utility may not show this difference clearly because it only plots over the valid parts of the domain and does not specify which points are not included in the domain.

Step by step solution

01

Determining the domain of function \(f(x)\)

To determine which values of x make the denominator 0, set \(x^2 - 2x = 0\). Solving this gives the roots x = 0 and x = 2. It means these x-values make the denominator 0 which would lead to undefined values. Thus, the domain of \(f(x)\) is all real numbers except 0 and 2.
02

Determining the domain of function \(g(x)\)

For \(g(x)=\frac{1}{x}\), the domain would be all x values that do not result in division by zero. So, all real numbers except 0 are in the domain of the function \(g(x)\).
03

Graphing the functions

Use a graphing utility to graph \(f(x)\) and \(g(x)\). This helps in visualizing how the functions look.
04

Analyzing the graphs

The graphing utility shows the overall shape of the functions and visually represents their behavior. However, it may not accurately represent the difference in the domains of \(f(x)\) and \(g(x)\). This is because the graphing utility only graphs over the valid parts of the domain, and places vertical asymptotes where the function is undefined, but does not specifically highlight which points in the domain make the function undefined.

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