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Chapter 1: Problem 37

Determine the domain of the function represented by the given equation. $$f(x)=\frac{1}{\sqrt{x+4}}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\frac{1}{\sqrt{x+4}}\) is \(( -4, +\infty)\).

Step by step solution

01

Identify the denominator and the radicand

In this function, the denominator is \(\sqrt{x+4}\). Also, the radicand, that is the expression inside the square root, is \(x+4\).
02

Set conditions for the denominator and the radicand

To make sure that the function is defined, the denominator should not be zero and the radicand should be non-negative. Thus, set the following conditions: \(\sqrt{x+4}\) not equal to zero, meaning \(x+4 > 0\), and \(x+4 \geq 0\) for square root to be real. Both conditions imply \(x > -4\).
03

Determine the domain

The solutions to both conditions from step 2 provide the domain of the function. Hence, the function is defined for all real numbers which are greater than -4. It means the domain of \(f(x)=\frac{1}{\sqrt{x+4}}\) is \(( -4, +\infty)\).

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