Question

The graph of the piecewise function shown. What is the range of f(x)?

Answer 1

$\underline{\text{piece-wise function}}$ :- actually , piece- wise function means, a function which is not continuous or non differential at some points belongs to its domain. example :- y = |x| is non- differential at x = 0 , so y = |x| is known as $\underline{\text{piece-wise function}}$ .
some importants $\underline{\text{piece-wise function}}$ are :
1.$\underline{\text{modulus function}}$
2.$\underline{\text{Signum function}}$
3. $\underline{\text{greatest integer function}}$
4. $\underline{\text{least integer function}}$ etc ...

how to find range :-
for finding range , there are lot of methods. But for piece wise fiction. You should use $\underline{\bold{Graph}}$. yes, you should plot the graph of piece -wise function. if y = f(x) is a piece wise function. Then range is all defined set of y.

If function is so lengthy. Then you should try to solve by using all given conditions . Example :- if given y = sign(x^2 + x + 2) . This is little tough to draw graph for it .
So, just apply conditions
We know, sign(x) = { 1 , when x > 0
= { -1 , when x < 0
= { 0 , when x = 0
∴ y = sign{x² + x + 2) = {1 , when x² + x + 2 > 0
= Ф , because x² + x + 2 can't be negative .
hence, range of y = sign{x² + x + 2} is { 1}

Now , I hope you got the skill for solving questions.