Find the domain of the function $$f(x) = \frac{1}{2x+1/x}.$$ Express your answer in interval notation.
The expression $\frac{1}{2x + \frac{1}{x}}$ is defined as long as the denominator is not zero. The denominator is zero when $2x + \frac{1}{x} = 0.$ This equation simplifies to $2x^2 + 1 = 0.$ Since $x^2$ is always nonnegative, there are no real solutions to this equation. Therefore, $2x + \frac{1}{x}$ is never equal to zero for any real number $x$, so the function $f(x) = \frac{1}{2x + \frac{1}{x}}$ is defined for all real numbers $x.$ In interval notation, the domain is $\boxed{(-\infty,\infty)}.$
To find the domain of a function, we need to identify all values of $x$ for which the function is defined. In this case, the function $f(x)$ is defined as long as the denominator $2x + \frac{1}{x}$ is not equal to zero. Therefore, we need to solve the equation $2x + \frac{1}{x} \neq 0$.
Multiplying both sides of the equation by $x$, we get $2x^2 + 1 \neq 0$. Subtracting $1$ from both sides, we have $2x^2 \neq -1$. Dividing both sides by $2$, we obtain $x^2 \neq -\frac{1}{2}$.
To solve this inequality, we take the square root of both sides. However, since the variable $x$ is in the denominator of the initial function, we must exclude the value $x=0$ from the domain.
Taking the square root of both sides yields $|x| \neq \sqrt{-\frac{1}{2}}$. Since the square root of a negative number does not exist in the real number system, the equation has no real solutions. Therefore, there are no values of $x$ that make the function undefined.
Hence, the domain of the function $f(x)$ is the set of all real numbers, expressed in interval notation as $(-\infty, \infty)$.
To find the domain of the function, we need to consider the values of $x$ that make the function well-defined. In this case, we have a rational function with a denominator $2x + \frac{1}{x}$.
To ensure that the denominator is not equal to zero, we need to exclude any values of $x$ that make the denominator zero. So, we set the denominator equal to zero and solve for $x$:
$$2x + \frac{1}{x} = 0$$
To solve this equation, we can multiply through by $x$ to get rid of the fraction:
$$2x^2 + 1 = 0$$
Rearranging this equation, we have:
$$2x^2 = -1$$
Dividing both sides by 2 gives:
$$x^2 = -\frac{1}{2}$$
Since the equation $x^2 = -\frac{1}{2}$ has no real solutions, it means that there are no values of $x$ that make the denominator zero. Therefore, there are no restrictions on the domain of the function.
Hence, the domain of the function $f(x) = \frac{1}{2x + \frac{1}{x}}$ is all real numbers. In interval notation, we can express this as $(-\infty, \infty)$.