Find the Domain and Range of f(x)=-|x|

Find the Domain and Range of $f(x)=-|x|$

Question: Find the domain and range of the real valued function: $$ f(x)=-|x| $$

Solution

The modulus function is defined for every real number.

Hence, the domain is: $$ \mathbb{R} $$

Since $$ |x|\ge0 $$

Therefore, $$ -|x|\le0 $$

Maximum value occurs at $$ x=0 $$

$$ f(0)=-|0|=0 $$

As $|x|$ increases, $-|x|$ decreases without bound.

Hence, the range is: $$ (-\infty,0] $$

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