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Chapter 2: Problem 63

Find the domain of each function. $$f(x)=\sqrt{\frac{2 x}{x+1}-1}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\sqrt{\frac{2 x}{x+1}-1}\) is all real \(x\) that satisfy both \(x+1 鈮� 0\) and \(\frac{2x}{x+1}-1 鈮� 0\), excluding \(x=-1\).

Step by step solution

01

- Identify Restrictions from the Denominator

Identify that the denominator of the fraction cannot be equal to 0. In this case, that means \(x+1 鈮� 0\). So, \(x 鈮� -1\). This is the first restriction for the domain.
02

- Set the Radicand Greater Than or Equal to 0

Set \(\frac{2x}{x+1}-1 鈮� 0\) and solve for \(x\). This involves clearing the fraction, combining like terms, and solving the resulting quadratic equation. The solutions to this form the second part of the domain.
03

- Combine the Restrictions

Combine the restrictions from Step 1 and Step 2 to determine the overall domain of the function. The domain is all real \(x\) which satisfy both conditions. Hence, the domain will be the intersection of the solutions got from step 1 and step 2.

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