# Quick question on simplifying expressions and functions of absolute values

I was wondering when simplifying an expression AND function, would I have to include the "restriction/domain" following the simplified expression and function, such as when there is a hole or something that cancels out? (See below to understand what I mean.)

Question: Simplify $$\frac{|x^3 + x|}{x}$$, when x<0.

$$\frac{|x^3 + x|}{x}$$

= |x^2 + 1|(-x) / x

$$= -|x^2 +1|$$

$$= -x^2 - 1, x ≠ 0$$

What I am confused about/what I want to know:

Do I need to include the $$x ≠ 0$$ with the simplified expression, or just leave it as $$-x^2 - 1$$?

ALSO, if I were then to restate this expression as a function of x, f(x), would I then state the $$x ≠ 0$$, following the simplified function? For example, if I was told to graph the simplified function, would I label my answer as $$f(x) = -x^2 - 1, x ≠ 0$$?

(I ask this because I remember doing something similar when dealing with finding vertical asymptotes of a function. If they cancel out, you would have to indicate something after the answer, like x can not be equal to this value)

• If the question is to simplify the expression when $x<0$ you don't have to worry about $x$ being equal to $0$. – Kabo Murphy Sep 12 at 5:54

Let's think about what the function is. The absolute value function is defined as $$|x^3+x|=\begin{cases}x^3+x,\text{ if }x^3+x\ge 0\\ -x^3-x,\text{ if }x^3+x<0. \end{cases}$$

Here, the function is defined only for $$x<0$$. Notice that $$x^3+x=x(x^2+1)$$ is zero only when $$x=0$$. You can check that $$x^3+x$$ is negative to the left of $$0$$, so the function is actually

$$f(x)=\frac{-x^3-x}{x}$$

on its domain. Since $$0$$ is not in the domain, we may assume $$x\neq 0$$, and so

$$f(x)=-x^2-1.$$

Just to be clear, the following statements are correct:

(1) When $$x<0$$, the expression $$\frac{|x^3+x|}{x}$$ is equal to $$-x^2-1$$.

(2) When $$x>0$$, the expression $$\frac{|x^3+x|}{x}$$ is equal to $$x^2+1$$.

(3) When $$x=0$$, the expression $$\frac{|x^3+x|}{x}$$ is undefined.

Note that an expression is not a function, so if you want to be precise, we never talk about the "domain" or "range" of an expression.

Now for a function $$f(x)=\frac{|x^3+x|}{x}$$, you need to know the domain of $$f$$ in order to make any of these statements. Generally, if the domain is not given, we assume it to be the largest possible domain which makes sense. In this case, I would assume the domain is all real numbers except $$0$$. Thus, we could write

$$f(x)=\begin{cases}-x^2-1,\text{ if }x<0\\ x^2+1,\text{ if }x>0. \end{cases}$$

• Thank you so much!!! Very clear explanation – herick the math leaner Sep 12 at 6:38