To determine the domain and range of the function [tex]\( f(x) = |x - 3| + 6 \)[/tex], we can break down the problem step-by-step.
1. Domain:
- The function [tex]\( f(x) = |x - 3| + 6 \)[/tex] consists of an absolute value function and a constant addition.
- The absolute value function [tex]\( |x - 3| \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- Adding a constant (in this case, 6) to a function does not affect the domain.
- Hence, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
Domain: [tex]\(\{x \mid x \text{ is all real numbers}\}\)[/tex]
2. Range:
- The absolute value function [tex]\( |x - 3| \)[/tex] has its minimum value at 0 (which occurs when [tex]\( x = 3 \)[/tex]).
- Therefore, the minimum value of [tex]\( f(x) = |x - 3| + 6 \)[/tex] will be [tex]\( 0 + 6 = 6 \)[/tex].
- As [tex]\( x \)[/tex] moves away from 3 in either direction (positive or negative), the value of [tex]\( f(x) \)[/tex] increases without bound.
- This means the function can take any value greater than or equal to 6.
Range: [tex]\(\{y \mid y \geq 6\}\)[/tex]
Given these steps, we can select the correct option from the provided choices:
- Domain: [tex]\(\{x \mid x \text{ is all real numbers}\}\)[/tex]
- Range: [tex]\(\{y \mid y \geq 6\}\)[/tex]
Thus, the correct option is:
Domain: [tex]\(\{x \mid x \text{ is all real numbers}\}\)[/tex]
Range: [tex]\(\{y \mid y \geq 6\}\)[/tex]
