Problem

The key is to recognize that this function's behavior splits into two cases based on the sign of $x$, and then use symmetry to understand its full range. For $x \geq 0$, the absolute value simplifies nicely; for $x < 0$, you get a different expression. Once you simplify both pieces, check whether $f$ is odd or even—this symmetry will tell you that if one side covers $[0, a)$, the other covers $(-a, 0]$, giving you the complete range without evaluating limits separately.

The key is to recognize that this function's behavior splits into two cases based on the sign of $x$, and then use symmetry to understand its full range. For $x \geq 0$, the absolute value simplifies nicely; for $x < 0$, you get a different expression. Once you simplify both pieces, check whether $f$ is odd or even—this symmetry will tell you that if one side covers $[0, a)$, the other covers $(-a, 0]$, giving you the complete range without evaluating limits separately.