Problem 66
Question
Find the domain of the function. $$f(x)=\frac{\sqrt{x+6}}{6+x}$$
Step-by-Step Solution
Verified Answer
The domain of the function \( f(x) = \frac{\sqrt{x+6}}{6+x} \) is \(x > -6\).
1Step 1: Find the Condition for the Square Root
The square root must have a non-negative argument. So \(x+6 >= 0\) which implies that \(x >= -6\). So, all real numbers greater than or equal to -6 will not make the square root undefined.
2Step 2: Find the Condition for the Denominator
The denominator of the fraction cannot equal zero because division by zero is undefined. So, \(6+x != 0\) which simplifies to \(x != -6\). Thus, -6 should be excluded from the domain.
3Step 3: Combine the Conditions
The conditions derived from the square root and the denominator need to be combined. This means that we are looking for all real numbers which are greater than or equal to -6, but not equal to -6. The domain of the function, therefore, is \(x > -6\).
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