To determine the range of the function [tex]\( f(x) = -|x| - 3 \)[/tex], let's analyze its behavior step by step.
1. Understanding [tex]\( |x| \)[/tex]:
- The notation [tex]\( |x| \)[/tex] represents the absolute value of [tex]\( x \)[/tex], which is always non-negative. That is, [tex]\( |x| \geq 0 \)[/tex] for any real number [tex]\( x \)[/tex].
2. Implications for [tex]\( -|x| \)[/tex]:
- Since [tex]\( |x| \geq 0 \)[/tex], the expression [tex]\( -|x| \)[/tex] will be less than or equal to 0 because negating a non-negative number results in a non-positive number. Therefore, [tex]\( -|x| \leq 0 \)[/tex].
3. Combining with the constant:
- The function [tex]\( f(x) \)[/tex] adds [tex]\(-|x|\)[/tex] to [tex]\(-3\)[/tex]. So we have [tex]\( f(x) = -|x| - 3 \)[/tex].
- Since [tex]\( -|x| \leq 0 \)[/tex], adding [tex]\(-3\)[/tex] means we are subtracting 3 from a non-positive number.
4. Finding the minimum value:
- For [tex]\( x = 0 \)[/tex], [tex]\( |x| = 0 \)[/tex]. Hence, [tex]\( f(0) = -|0| - 3 = -3 \)[/tex].
- This is the maximum value of [tex]\( f(x) \)[/tex] because for any other [tex]\( x \)[/tex], [tex]\( |x| \)[/tex] will be greater than 0, making [tex]\( -|x| \)[/tex] more negative and thus making [tex]\( f(x) \)[/tex] less than [tex]\(-3\)[/tex].
5. Determining the range:
- As [tex]\(|x|\)[/tex] increases from 0 to positive infinity, [tex]\( -|x| \)[/tex] will go from 0 to negative infinity, and hence [tex]\( f(x) = -|x| - 3 \)[/tex] will go from [tex]\(-3\)[/tex] towards negative infinity.
- Therefore, [tex]\( f(x) \)[/tex] can take any value that is less than or equal to [tex]\(-3\)[/tex].
In conclusion, the range of the function [tex]\( f(x) = -|x| - 3 \)[/tex] is:
All real numbers less than or equal to [tex]\(-3\)[/tex].
