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Chapter 2: Problem 6

Find the domain of the function and discuss the behavior of \(f\) near any excluded \(x\) -values. $$f(x)=\frac{5 x}{x+2}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\frac{5x}{x+2}\) is all real numbers except \(x=-2\) and can be written in interval notation as \((-鈭�,-2)鈭�(-2,鈭�)\). The function has a vertical asymptote at \(x=-2\), approaching \(鈭瀄) from the left and \(-鈭瀄) from the right.

Step by step solution

01

Find the Excluded x-value

Set the denominator equal to zero and solve for \(x\) to find the excluded value:\nThe denominator is \(x+2\)\nSo, solve the equation \(x+2=0\)\nThis results in \(x=-2\)\nSo, \(x=-2\) is the value that makes the denominator zero and thus, the function undefined.
02

Determine the Domain of the Function

The domain of the function is all real numbers except \(x=-2\). This can be written in interval notation as \((-鈭�,-2)鈭�(-2,鈭�)\).
03

Discuss the Behavior of f near the Excluded x-value

To better understand the behavior of \(f\) as \(x\) approaches -2, we will examine the function as \(x\) approaches -2 from the left and from the right.\n\nAs \(x\) approaches -2 from the left (\(x鈫�-2^-)\), the function approaches \(鈭瀄). This can be seen by substituting a number slightly less than -2 into the function, such as -2.01.\n\nAs \(x\) approaches -2 from the right (\(x鈫�-2^+)\), the function approaches \(-鈭瀄). This can be seen by substituting a number slightly greater than -2 into the function, such as -1.99.\n\nTherefore, the function has a vertical asymptote at \(x=-2\).

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