Find the domain of y = log10[sin(x-3) + ​√(16-x2) ] ( where 10 is base)

Question

y = log₁₀[sin(x-3) + √(16- x²)]

Answer 1

The argument of the log has certain conditions to be met.

 

1)  No even radicals can contain a negative value in it.

2)  The argument cannot be negative nor zero.

 

16 - x² > 0

 

For the second condition, we know the domain is (-4, 4).  Because of this condition, the domain of the whole function is (-4, 4).

 

Sine does not affect the domain, even though we know sine ranges from -1 to 1.  However, -1 added to a positive number is still positive.  Also, adding 1 to a positive number is also positive.  This keeps the argument of log positive.