I'm stuck on this problem:
$$f(x) = \frac{x^2 -4}{x}$$
I need to determine why this function's domain is not: $$\{x|x \neq \pm 2\}$$ All of the examples that I've seen have the quadratic in the denominator, and then you just factor them out, and solve for x. How do you find the domain of this function algebraically?
The only value not allowed is one that will make the denominator equal to $0$, because you cannot divide by $0$. In this case, setting $x$ to $0$ will make the denominator equal to $0$. So, $x\neq 0$. Therefore the domain is: $$\{x\mid x\neq 0\}$$
Extra details:
The graph of the function is:
Notice how there is a vertical asymptote at $x=0$. The graph starts approaching infinity at the left, reaches $x=0$ where it is not defined, and then "drops" all the way to negative infinity and starts increasing again.
Why the domain is not $\{x\mid x\neq \pm 2\}$:
Remember, the values not allowed come are ones that make the denominator equal $0$, not the numerator. It seems that you are trying to find the $x$-intercepts. To find the $x$-intercepts, one must make $f(x)=0$ . In this case, you would make the numerator equal to $0$. That means solving for $x$ in the equation $x^2-4=0$, which gives you $x=\pm 2$.