To identify the domain of the combined functions \((f + g)(x)\), \((f - g)(x)\), \((f \cdot g)(x)\), and \((f / g)(x)\), we will consider the individual domains of \(f(x)\) and \(g(x)\) and any restrictions that arise from their operations. Below is a general explanation without specific functions, as you did not provide the definitions of \(f(x)\) and \(g(x)\).
Domain of Combined Functions
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Domain of \((f + g)(x)\): \[ \text{Domain} = { x ,|, x \in \text{Domain of } f \text{ and } x \in \text{Domain of } g } \]
- Restrictions: The domain of \((f + g)(x)\) includes only values of \(x\) that are in both the domain of \(f\) and the domain of \(g\).
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Domain of \((f - g)(x)\): \[ \text{Domain} = { x ,|, x \in \text{Domain of } f \text{ and } x \in \text{Domain of } g } \]
- Restrictions: Similar to addition, the domain of \((f - g)(x)\) includes values of \(x\) in both domains.
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Domain of \((f \cdot g)(x)\): \[ \text{Domain} = { x ,|, x \in \text{Domain of } f \text{ and } x \in \text{Domain of } g } \]
- Restrictions: The domain of \((f \cdot g)(x)\) involves values of \(x\) that exist in both domains.
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Domain of \((f / g)(x)\): \[ \text{Domain} = { x ,|, x \in \text{Domain of } f \text{ and } x \in \text{Domain of } g , \text{and} , g(x) \neq 0 } \]
- Restrictions: The domain of \((f / g)(x)\) includes values of \(x\) in both domains, with the additional condition that \(g(x) \neq 0\) to avoid division by zero.
Summary
- For all functions except \((f / g)(x)\), the domain is the intersection of the domains of \(f\) and \(g\).
- For \((f / g)(x)\), the domain is the intersection of the domains of \(f\) and \(g\) with the restriction that \(g(x) \neq 0\).
You can copy and paste this text as needed. If you have specific forms of \(f(x)\) and \(g(x)\), you could apply these general principles to determine specific domains.