To determine the range of the function [tex]\( f(x) = -2|x+1| \)[/tex], we need to analyze how the values of the function change with respect to [tex]\( x \)[/tex].
1. Understanding the Absolute Value Function:
- The expression [tex]\( |x+1| \)[/tex] denotes the absolute value of [tex]\( x+1 \)[/tex].
- The absolute value function [tex]\( |x+1| \)[/tex] reaches its minimum value when [tex]\( x+1 = 0 \)[/tex].
- So, the minimum value of [tex]\( |x+1| \)[/tex] is 0 when [tex]\( x = -1 \)[/tex].
2. Applying the Transformation:
- The function [tex]\( f(x) = -2|x+1| \)[/tex] takes the absolute value [tex]\( |x+1| \)[/tex] and multiplies it by [tex]\(-2\)[/tex].
- When [tex]\( x = -1 \)[/tex], [tex]\( |x+1| = 0 \)[/tex], thus:
[tex]\[ f(-1) = -2 \cdot 0 = 0 \][/tex]
3. Behavior for Other Values of [tex]\( x \)[/tex]:
- For any [tex]\( x \)[/tex] other than [tex]\(-1\)[/tex], [tex]\( |x+1| \)[/tex] is positive.
- Since we are multiplying by [tex]\(-2\)[/tex], the result will be negative or zero.
4. Finding the Range:
- As [tex]\( x \)[/tex] moves away from [tex]\(-1\)[/tex], the value of [tex]\( |x+1| \)[/tex] increases, making [tex]\(-2|x+1|\)[/tex] more negative.
- Therefore, the function [tex]\( f(x) = -2|x+1| \)[/tex]:
- Has a maximum value of 0 when [tex]\( x = -1 \)[/tex].
- Becomes smaller than 0 (i.e., negative) as [tex]\( x \)[/tex] deviates from [tex]\(-1\)[/tex].
5. Conclusion:
- The range of [tex]\( f(x) = -2|x+1| \)[/tex] includes all real numbers that are less than or equal to 0.
Thus, the correct choice for the range of the function is:
- All real numbers less than or equal to 0.
