Bot GPT 4-o mini To determine the restrictions on the domain of the expression involving \( u(x) \) and \( v(x) \), we first need to analyze the provided domain restrictions for each function and how they interact.
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Domain of \( u(x) \): The function \( u(x) \) is defined for all real values except \( 0 \). Therefore, \( x \neq 0 \).
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Domain of \( v(x) \): The function \( v(x) \) is defined for all real values except \( 2 \). Therefore, \( x \neq 2 \).
Next, if \( u(x) \) and \( v(x) \) are involved in some combined operation, like addition, subtraction, multiplication, or division, we need to consider the combined restrictions.
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If we are adding or subtracting \( u(x) \) and \( v(x) \), the domain will be the intersection of the domains of both functions. Thus, the restrictions are:
- \( x \neq 0 \)
- \( x \neq 2 \)
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If we are multiplying \( u(x) \) and \( v(x) \), the same restrictions apply:
- \( x \neq 0 \)
- \( x \neq 2 \)
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If we are dividing \( u(x) \) by \( v(x) \) or vice versa, we need to ensure that the denominator is not zero. Since \( v(x) \) is defined for all values except \( 2 \), \( v(x) \) will not affect the numerator \( u(x) \) as \( u(x) \) does not equal or depend on \( 2 \) based on restrictions. Therefore, the same conditions apply:
- \( x \neq 0 \)
- \( x \neq 2 \)
Overall, the restrictions on the domain based on the conditions you provided will be:
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The domain of the combined expression involving \( u(x) \) and \( v(x) \) is all real numbers except \( 0 \) and \( 2 \). This can be expressed as:
\[ x \neq 0 \quad \text{and} \quad x \neq 2 \]
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