To determine the range of the function [tex]\( f(x) = -|x| - 3 \)[/tex], we need to analyze how the function behaves and what values it can take.
1. Understanding the Absolute Value:
The absolute value of [tex]\( x \)[/tex], denoted [tex]\( |x| \)[/tex], is always non-negative. This means [tex]\( |x| \geq 0 \)[/tex] for any [tex]\( x \)[/tex].
2. Effect of the Negative Sign:
The function [tex]\( -|x| \)[/tex] will negate the absolute value. Hence, [tex]\( -|x| \)[/tex] is always non-positive, [tex]\( -|x| \leq 0 \)[/tex].
3. Shifting the Function Down:
The function [tex]\( f(x) = -|x| - 3 \)[/tex] can be seen as shifting the graph of [tex]\( -|x| \)[/tex] downward by 3 units. Therefore, any value that [tex]\( -|x| \)[/tex] takes, when reduced by 3, will be further decreased.
4. Determining the Function's Range:
Considering the range of [tex]\( -|x| \leq 0 \)[/tex]:
- When you subtract 3, the smallest possible value [tex]\( -|x| \)[/tex] can take is further reduced:
[tex]\[
f(x) = -|x| - 3 \implies -|x| - 3 \leq -3
\][/tex]
This means the values of [tex]\( f(x) \)[/tex] are all less than or equal to -3.
Therefore, the range of [tex]\( f(x) = -|x| - 3 \)[/tex] is all real numbers less than or equal to -3.
Thus, the correct answer is: all real numbers less than or equal to -3.
