Answer :
To find the range of the function [tex]\( g(x) = |x - 6| - 8 \)[/tex], let's analyze the behavior of the function step by step.
1. Understanding the Absolute Value Function:
The function [tex]\( g(x) \)[/tex] involves the absolute value expression [tex]\( |x - 6| \)[/tex]. The absolute value function [tex]\( |x| \)[/tex] returns the non-negative value (distance from zero) of [tex]\( x \)[/tex]. Therefore, the expression [tex]\( |x - 6| \)[/tex] represents the distance of [tex]\( x \)[/tex] from 6, which is always non-negative. This means:
[tex]\[ |x - 6| \geq 0 \quad \text{for all } x. \][/tex]
2. Shifting the Absolute Value Function:
The function [tex]\( g(x) = |x - 6| - 8 \)[/tex] subtracts 8 from [tex]\( |x - 6| \)[/tex]. We know that [tex]\( |x - 6| \)[/tex] can take on all non-negative values starting from 0. So, the minimum value of [tex]\( |x - 6| \)[/tex] is 0, which occurs when [tex]\( x = 6 \)[/tex].
3. Determining the Minimum Value of [tex]\( g(x) \)[/tex]:
When [tex]\( x = 6 \)[/tex]:
[tex]\[ g(6) = |6 - 6| - 8 = 0 - 8 = -8. \][/tex]
This shows that the minimum value of [tex]\( g(x) \)[/tex] is -8.
4. Range of [tex]\( g(x) \)[/tex]:
As [tex]\( |x - 6| \)[/tex] increases from 0 upwards without bound (for values of [tex]\( x \)[/tex] other than 6), [tex]\( g(x) \)[/tex] will take on values:
[tex]\[ g(x) = |x - 6| - 8 \geq 0 - 8 = -8. \][/tex]
This means that [tex]\( g(x) \)[/tex] can be equal to -8 or any value greater than -8. Therefore, the range of [tex]\( g(x) \)[/tex] includes all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq -8 \)[/tex].
Putting this all together, the range of the function [tex]\( g(x) = |x - 6| - 8 \)[/tex] is:
[tex]\[ \{ y \mid y \geq -8 \}. \][/tex]
Hence, the correct choice is:
\[
\{ y \mid y \geq -8 \}.
\
1. Understanding the Absolute Value Function:
The function [tex]\( g(x) \)[/tex] involves the absolute value expression [tex]\( |x - 6| \)[/tex]. The absolute value function [tex]\( |x| \)[/tex] returns the non-negative value (distance from zero) of [tex]\( x \)[/tex]. Therefore, the expression [tex]\( |x - 6| \)[/tex] represents the distance of [tex]\( x \)[/tex] from 6, which is always non-negative. This means:
[tex]\[ |x - 6| \geq 0 \quad \text{for all } x. \][/tex]
2. Shifting the Absolute Value Function:
The function [tex]\( g(x) = |x - 6| - 8 \)[/tex] subtracts 8 from [tex]\( |x - 6| \)[/tex]. We know that [tex]\( |x - 6| \)[/tex] can take on all non-negative values starting from 0. So, the minimum value of [tex]\( |x - 6| \)[/tex] is 0, which occurs when [tex]\( x = 6 \)[/tex].
3. Determining the Minimum Value of [tex]\( g(x) \)[/tex]:
When [tex]\( x = 6 \)[/tex]:
[tex]\[ g(6) = |6 - 6| - 8 = 0 - 8 = -8. \][/tex]
This shows that the minimum value of [tex]\( g(x) \)[/tex] is -8.
4. Range of [tex]\( g(x) \)[/tex]:
As [tex]\( |x - 6| \)[/tex] increases from 0 upwards without bound (for values of [tex]\( x \)[/tex] other than 6), [tex]\( g(x) \)[/tex] will take on values:
[tex]\[ g(x) = |x - 6| - 8 \geq 0 - 8 = -8. \][/tex]
This means that [tex]\( g(x) \)[/tex] can be equal to -8 or any value greater than -8. Therefore, the range of [tex]\( g(x) \)[/tex] includes all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq -8 \)[/tex].
Putting this all together, the range of the function [tex]\( g(x) = |x - 6| - 8 \)[/tex] is:
[tex]\[ \{ y \mid y \geq -8 \}. \][/tex]
Hence, the correct choice is:
\[
\{ y \mid y \geq -8 \}.
\
