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Chapter 0: Problem 30

Find the domain of the function. $$ f(x)=\frac{x}{2+\sin x} $$

Short Answer

Expert verified
The domain of the function \(f(x) = \frac{x}{2 + \sin x}\) is all real numbers, since there is no value of \(x\) that makes the denominator zero. In interval notation, the domain is \((- \infty, \infty)\) or in set-builder notation as \({x: x \in \mathbb{R}}\).

Step by step solution

01

Set the denominator equal to zero

In this step, we will set the denominator of the function equal to zero and solve for \(x\). This will allow us to find out where the function is undefined (i.e., where the denominator is zero since division by zero is undefined). So, we first set the denominator equal to zero: \(2 + \sin x = 0\)
02

Solve for x

Now, try to isolate \( \sin x \) on one side of the equation: \( \sin x = -2 \) But, the range of the sine function is \(-1 \leq \sin x \leq 1\). That means there is no value of \(x\) that makes \(\sin(x) = -2\) true.
03

Determine the domain

There is no need to exclude any values from the domain based on the above steps, as there can't be \(x\) values for which the denominator becomes zero (step 2). Therefore, the domain of the function \(f(x) = \frac{x}{2 + \sin x}\) is all real numbers. You can write it in interval notation as: \((- \infty, \infty)\) or in set-builder notation as \({x: x \in \mathbb{R}}\).

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