To determine the domain of the function
y=+√x+7, we need to consider the constraints imposed by both components of the function:
The term
requires
log10(3−x) to be a defined, non-zero value. The logarithm
log10(3−x) is defined when
3−x is positive, so:
3−x>0⇒x<3Also,
log10(3−x)≠0, which implies:
log10(3−x)=0⇒3−x=1⇒x=2Thus,
x≠2.
The term
√x+7 requires the argument inside the square root to be non-negative:
x+7≥0⇒x≥−7 Combining these constraints, we get:
−7≤x<3x≠2Thus, the domain of the function is the interval
[−7,3) excluding
x=2. This corresponds to Option C .
So, the correct answer is:
Option C
[−7,3)−{2}