Question

find the domain of the function. f(x) = \frac{x}{8x^{2}+5}. the domain is . (type your answer in interval notation.)

Explanation:

Step1: Recall domain - finding rule

The domain of a rational function $\frac{f(x)}{g(x)}$ is all real numbers except where $g(x)=0$. Here, $f(x) = x$ and $g(x)=8x^{2}+5$.

Step2: Set the denominator equal to zero

We need to solve the equation $8x^{2}+5 = 0$. Rearrange it to $8x^{2}=-5$, then $x^{2}=-\frac{5}{8}$.

Step3: Analyze the solution of the equation

Since the square of any real - number $x$, i.e., $x^{2}\geq0$ for all real $x$, the equation $x^{2}=-\frac{5}{8}$ has no real solutions.

Answer:

$(-\infty,\infty)$