To determine the domain and range of the function [tex]\( f(x) = |x+6| \)[/tex]:
1. Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept. The function [tex]\( f(x) = |x + 6| \)[/tex] involves an absolute value, which means that you can input any real number into [tex]\( x \)[/tex] since the absolute value function is defined for all real numbers.
Therefore, the domain of [tex]\( f(x) = |x + 6| \)[/tex] is all real numbers, which can be represented as:
[tex]\[
(-\infty, \infty)
\][/tex]
2. Range:
The range of a function is the set of all possible output values (y-values) that the function can produce. For [tex]\( f(x) = |x + 6| \)[/tex], the absolute value of any number is always non-negative. This implies that the output of [tex]\( f(x) \)[/tex] will always be greater than or equal to 0.
Therefore, the range of [tex]\( f(x) = |x + 6| \)[/tex] is all non-negative real numbers, which can be represented as:
[tex]\[
[0, \infty)
\][/tex]
Given these observations, let’s match our findings with the provided options:
- domain: [tex]\( (-\infty, \infty)\)[/tex]; range: [tex]\( f(x) \geq 0 \)[/tex]
- domain: [tex]\( x \leq-6\)[/tex]; range: [tex]\( (-\infty, \infty) \)[/tex]
- domain: [tex]\( x \geq-6\)[/tex]; range: [tex]\( (-\infty, \infty) \)[/tex]
- domain: [tex]\( (-\infty, \infty)\)[/tex]; range: [tex]\( f(x) \leq 0 \)[/tex]
The correct option is the first one:
- domain: [tex]\( (-\infty, \infty)\)[/tex]; range: [tex]\( f(x) \geq 0 \)[/tex]
