To solve for the domain and range of the function [tex]\( f(x) = |x-3| + 6 \)[/tex], we need to analyze the properties of the absolute value function and how it affects the given expression.
1. Domain:
The domain of [tex]\( f(x) \)[/tex] refers to all the possible values of [tex]\( x \)[/tex] for which the function is defined. Since the function [tex]\( f(x) = |x-3| + 6 \)[/tex] involves the absolute value of [tex]\( x-3 \)[/tex], and the absolute value function is defined for all real numbers, there are no restrictions on [tex]\( x \)[/tex]. Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \{ x \mid x \text{ is all real numbers} \} \][/tex]
2. Range:
The range of [tex]\( f(x) \)[/tex] refers to all the possible values of [tex]\( y \)[/tex] that the function can take. To determine the range, we need to analyze the output values of [tex]\( f(x) \)[/tex].
Consider [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = |x-3| + 6 \][/tex]
The absolute value function [tex]\( |x-3| \)[/tex] is always non-negative, meaning [tex]\( |x-3| \geq 0 \)[/tex] for any real number [tex]\( x \)[/tex]. Therefore,
[tex]\[ |x-3| + 6 \geq 0 + 6 \][/tex]
[tex]\[ |x-3| + 6 \geq 6 \][/tex]
This shows that the minimum value of [tex]\( f(x) \)[/tex] is 6, which occurs when [tex]\( |x-3| = 0 \)[/tex] (i.e., when [tex]\( x = 3 \)[/tex]). For all other values of [tex]\( x \)[/tex], [tex]\( |x-3| \)[/tex] will be positive, making [tex]\( f(x) \)[/tex] greater than 6. Hence, the function [tex]\( f(x) \)[/tex] will take all values greater than or equal to 6.
Thus, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \{ y \mid y \geq 6 \} \][/tex]
Combining our findings:
- Domain: [tex]\(\{x \mid x \text{ is all real numbers} \}\)[/tex]
- Range: [tex]\(\{ y \mid y \geq 6 \}\)[/tex]
Hence, the correct answer is:
[tex]\[ \text{Domain: } \{ x \mid x \text{ is all real numbers} \} \][/tex]
[tex]\[ \text{Range: } \{ y \mid y \geq 6 \} \][/tex]
