Final answer:
The expression ( log_3(x+3) + log_2(x) ) is undefined for values of x where either of the arguments of the logarithmic functions is not positive. Since x must be greater than zero for log_2(x), and greater than -3 for log_3(x+3), the combined condition is that x must be greater than 0. Thus, Option D (x > -3) is the best answer provided.
Step-by-step explanation:
The original question asks for which values of x the logarithmic expression ( log_3(x+3) + log_2(x) ) is undefined. To determine when a logarithmic expression is undefined, we must look at the domain of logarithmic functions. The logarithm of a number is only defined for positive arguments. Therefore, the expression inside each logarithmic function must be greater than zero.
For the term log_3(x+3), the inside must be greater than zero: x + 3 > 0, hence x > -3. Similarly, for the term log_2(x), it requires that x > 0. The second restriction is the more restrictive one, eliminating options A and C and indicating that x must strictly be greater than zero. Thus options B and D need to be evaluated.
Considering both restrictions together, the combined requirement is that x must be greater than zero for the entire logarithmic expression to be defined. Option B (x < -3) does not satisfy the requirement for either logarithmic term, and Option D (x > -3) only satisfies the requirement for log_3(x+3). Only values of x that are strictly greater than 0 will make the whole expression defined, hence the correct answer must specify x > 0. While the options given don't include this exact inequality, it implies that Option D is the best answer as it allows for positive values of x as required by log_2(x).