To find the domain of the composite function \( (cd)(x) = c(x) \cdot d(x) \), we need to consider the domains of both functions \( c(x) \) and \( d(x) \).
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Function \( c(x) = \frac{5}{x - 2} \):
- The expression is undefined when the denominator equals zero, which happens when \( x - 2 = 0 \) or \( x = 2 \).
- Therefore, the domain of \( c(x) \) is all real numbers except \( x = 2 \).
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Function \( d(x) = x + 3 \):
- This is a linear function, which means it is defined for all real numbers.
- Therefore, the domain of \( d(x) \) is all real numbers.
Now, when we multiply \( c(x) \) and \( d(x) \) to form \( (cd)(x) = c(x) \cdot d(x) \), the overall domain will include restrictions from both functions.
Since \( c(x) \) is undefined only at \( x = 2 \) and \( d(x) \) is defined everywhere, the final domain of \( (cd)(x) \) will be all real numbers except \( x = 2 \).
Thus, the domain of \( (cd)(x) \) is:
All real values of x except x = 2.