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Chapter 1: Problem 58

Find the domain of the function. $$f(x)=\frac{\sqrt{x+6}}{6+x}$$

Short Answer

Expert verified
The domain of the function \(f(x) = \frac{\sqrt{x+6}}{x+6}\) is \(-6 < x \leq \infty\).

Step by step solution

01

Solve Equation for Square Root

Given that square roots only accept non-negative numbers, one must solve the equation \(x + 6 \geq 0\). This simplifies to \(x \geq -6\). This means that for the square root to work, the values of \(x\) should be \(-6\) or any number greater than \(-6\).
02

Identify Values that Would Result to Denominator Equals Zero and Exclude Them

We also know that a value in the denominator makes the denominator zero and makes the function undefined. The denominator \(6 + x\) equals 0 when \(x = -6\). Thus, the value \(-6\) makes the denominator zero and should be excluded from the function's domain.
03

The Domain of The Function

To summarize, based on the two previous steps, the domain for the function \(f(x) = \frac{\sqrt{x+6}}{x+6}\) is \(-6 < x \leq \infty\). This is achieved by considering all values greater than -6 (as realized from step 1) and excluding -6 which makes the denominator zero (as realized from step 2).

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