To determine the range of the function [tex]\( g(x) = |x - 12| - 2 \)[/tex], let's analyze it step by step.
1. Understanding the [tex]\( |x - 12| \)[/tex] term:
- The expression [tex]\( |x - 12| \)[/tex] represents the absolute value of the difference between [tex]\( x \)[/tex] and 12.
- Absolute values always yield non-negative results, meaning [tex]\( |x - 12| \geq 0 \)[/tex] for any real number [tex]\( x \)[/tex].
2. Minimum value of [tex]\( |x - 12| \)[/tex]:
- The minimum value of [tex]\( |x - 12| \)[/tex] is 0, which occurs when [tex]\( x = 12 \)[/tex].
3. Transforming the expression:
- Given that the minimum value of [tex]\( |x - 12| \)[/tex] is 0, substitute [tex]\( x = 12 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(12) = |12 - 12| - 2 = 0 - 2 = -2
\][/tex]
- This means the minimum value that [tex]\( g(x) \)[/tex] can take is [tex]\( -2 \)[/tex].
4. Behavior of the function:
- Since [tex]\( |x - 12| \geq 0 \)[/tex], the lowest value inside the absolute value term is 0, after which it only increases.
- As [tex]\( |x - 12| \)[/tex] increases from 0, adding higher positive numbers to -2 results in [tex]\( g(x) \)[/tex] increasing without any upper limit.
- Therefore, for all [tex]\( x \)[/tex], [tex]\( g(x) = |x - 12| - 2 \)[/tex] will yield values starting from [tex]\( -2 \)[/tex] and extending to infinity.
5. Defining the range:
- The range of [tex]\( g(x) \)[/tex] starts at [tex]\( -2 \)[/tex] and includes all values greater than [tex]\( -2 \)[/tex].
- This can be expressed in set notation as [tex]\( \{ y \mid y \geq -2 \} \)[/tex].
Therefore, the correct answer is:
[tex]\[
\{ y \mid y \geq -2 \}
\][/tex]
