Find the domain of the function $$f(x) = \frac{1}{2x+1/x}.$$ Express your answer in interval notation.
1 answer
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 $\boxed{(-\infty,\infty)}.$