To determine the range of the function [tex]\( g(x) = -\frac{1}{2} |x - 6| + 1 \)[/tex], we need to analyze its behavior over all possible values of [tex]\( x \)[/tex].
1. Consider the structure of the function:
The function includes an absolute value, [tex]\( |x - 6| \)[/tex], which is always non-negative, meaning [tex]\( |x - 6| \geq 0 \)[/tex].
2. Analyze the expression [tex]\( -\frac{1}{2} |x - 6| \)[/tex]:
Since [tex]\( |x - 6| \geq 0 \)[/tex], when multiplied by [tex]\(-\frac{1}{2}\)[/tex], the result will be non-positive (less than or equal to zero).
3. Simplify the analysis:
[tex]\[
-\frac{1}{2} |x - 6| \leq 0
\][/tex]
Because [tex]\( -\frac{1}{2} \)[/tex] is negative, and [tex]\( |x - 6| \)[/tex] is zero or positive. Now, adding 1 to this expression:
[tex]\[
-\frac{1}{2} |x - 6| + 1 \leq 1
\][/tex]
4. Determine the maximum value:
When [tex]\( |x - 6| = 0 \)[/tex] (which occurs when [tex]\( x = 6 \)[/tex]), the function reaches its maximum value:
[tex]\[
g(6) = -\frac{1}{2} |6 - 6| + 1 = -\frac{1}{2} \cdot 0 + 1 = 1
\][/tex]
5. Determine the behavior for increasing [tex]\( |x - 6| \)[/tex]:
As [tex]\( |x - 6| \)[/tex] increases (as [tex]\( x \)[/tex] moves away from 6 in either direction), the term [tex]\( -\frac{1}{2} |x - 6| \)[/tex] becomes more negative, causing [tex]\( g(x) \)[/tex] to decrease without bound.
6. Conclusion regarding the range:
[tex]\( g(x) \)[/tex] can take any value less than or equal to 1.
Therefore, the range of [tex]\( g(x) = -\frac{1}{2} |x - 6| + 1 \)[/tex] is all real numbers less than or equal to 1, which can be expressed in interval notation as:
[tex]\[
(-\infty, 1]
\][/tex]
Thus, the correct answer is:
A. [tex]\((- \infty, 1]\)[/tex]
