What is the range of this combined function?

The range is $(0,\infty) \cup (-\infty, -\frac 1 3]$. To see this write the range as $\{\frac 1 {t-3}: t \geq 0, t \neq 3\}$. Find $\{\frac 1 {t-3}:0 \leq t < 3\}$ and $\{\frac 1 {t-3}: 3 < t <\infty)\}$ separately. These can be written as $\{\frac 1 s:-3 \leq s < 0\}$ and $\{\frac 1 s: 0 < s <\infty)\}$. Can you compute the range now?