Question

If c (x) = StartFraction 5 Over x minus 2 EndFraction and d(x) = x + 3, what is the domain of (cd)(x)?
all real values of x
all real values of x except x = 2
all real values of x except x = –3
all real values of x except x = 2 and x = –3

Answer 1

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 \( d(x) \).

1. **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 \( x = 2 \).

2. **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.**