Image of an absolute value function

Question

I need to find the image in regards to this question.

What is the image of the function $f : \mathbb{R}\to\mathbb{R}$ given by

$f(x) = \frac{|x+3|}{|x|+3}$?

What I tried to do is to set $y =$ to $f(x)$ and then get $x$:

Let $y = f(x) = \frac{|x+3|}{|x|+3}$

$y = \frac{|x+3|}{|x|+3}$

$y(|x|+3) = |x+3|$

I don't know how to go from here as I don't know what to do/how to get rid of the absolute value.

Answer 1

$|x+3|\ge0$ and $|x|+3>0$, so $f(x)\ge0$.

$|x+3|\le |x|+3$, so $f(x)\le1$.

$f(-3)=0$ and $f(0)=1$, and $f$ is continuous.

So the image is $[0,1]$.

Answer 2

** Hint:** Note that $x=-3$ and $x=0$ are the places where something “important” happens to the absolute value expressions. Everywhere in between and beyond, you can replace a symbol like $|u|$ with either $u$ or $-u$, whichever makes the expression $u$ nonnegative.

So consider the cases $x\leq -3$, $-3<x\leq 0$, and $x>0$ separately.