Indicate the domain and the image of the following function:
$$$f(x)=\left\{\begin{array}{rcl} -1 & \mbox{ if } & x<-1 \\ 2x+1 & \mbox{ if } & -1\leq x < 2 \\ 2 & \mbox{ if } & x\geq 3\end{array}\right.$$$
Development:
We can find the domain of the first function from the intervals in its definition:
$$$Dom (f) = (-\infty,-1)\cup[-1,2)\cup[3,+\infty)=(-\infty,2)\cup[3,+\infty)$$$
To determine the image we can concentrate on the images of the different functions that compose the function, bearing in mind the domain where they are defined.
For $$x < -1$$ or $$x > 3$$ we have no problems since we know the valuation of the function in these intervals.
For the straight line between $$-1$$ and $$2$$, we calculate the valuation in the above mentioned points:
$$2x+1$$ in $$x =-1$$ values $$-1$$
$$2x +1$$ in $$x = 2$$ values $$5$$
Therefore $$Im (f) = [-1, 5)$$.
It is necessary to bear in mind that we will include the extreme points in the image depending on whether they are included or not in the definition of the function.
Solution:
$$Dom(f)=(-\infty,2)\cup[3,+\infty)$$, $$Im (f) = [-1, 5)$$