To determine the range of the function [tex]\( f(x) = -2|x+1| \)[/tex], let's analyze it step-by-step.
1. Understanding the absolute value function:
- The function [tex]\( |x+1| \)[/tex] represents the absolute value of [tex]\( x+1 \)[/tex]. The absolute value function, [tex]\( |a| \)[/tex], returns the non-negative value of [tex]\( a \)[/tex]. Therefore, [tex]\( |x+1| \geq 0 \)[/tex] for all real numbers [tex]\( x \)[/tex].
2. Analyzing [tex]\( -2|x+1| \)[/tex]:
- Since [tex]\( |x+1| \)[/tex] is always non-negative, multiplying it by [tex]\(-2\)[/tex] will always yield a non-positive value. In other words, [tex]\( -2|x+1| \leq 0 \)[/tex].
- The minimum value of [tex]\( |x+1| \)[/tex] occurs when [tex]\( x = -1 \)[/tex]. In this case, [tex]\( |x+1| = |(-1)+1| = 0 \)[/tex], and thus [tex]\( f(x) = -2 \cdot 0 = 0 \)[/tex].
3. Behavior of the function as distance from [tex]\(-1\)[/tex] increases:
- As [tex]\( x \)[/tex] moves away from [tex]\(-1\)[/tex] in either direction, [tex]\( |x+1| \)[/tex] increases. When [tex]\( |x+1| \)[/tex] increases, the value of [tex]\( -2|x+1| \)[/tex] becomes more negative, meaning [tex]\( f(x) \)[/tex] decreases.
4. Finding the range:
- The maximum value [tex]\( f(x) \)[/tex] can take is [tex]\( 0 \)[/tex] when [tex]\( x = -1 \)[/tex].
- There is no lower bound to the values [tex]\( f(x) \)[/tex] can take because [tex]\( |x+1| \)[/tex] can grow indefinitely, making [tex]\( -2|x+1| \)[/tex] indefinitely negative.
- Hence, the range of [tex]\( f(x) \)[/tex] includes [tex]\( 0 \)[/tex] and all negative values.
Therefore, the range of the function [tex]\( f(x) = -2|x+1| \)[/tex] is all real numbers less than or equal to 0.
So the correct answer is:
all real numbers less than or equal to 0.
