What are the domain and range of [tex][tex]$f(x)=|x+6|$[/tex][/tex]?

A. Domain: [tex]$(-\infty, \infty)$[/tex]; Range: [tex]$f(x) \geq 0$[/tex]

B. Domain: [tex]$x \leq -6$[/tex]; Range: [tex]$(-\infty, \infty)$[/tex]

C. Domain: [tex]$x \geq -6$[/tex]; Range: [tex]$(-\infty, \infty)$[/tex]

D. Domain: [tex]$(-\infty, \infty)$[/tex]; Range: [tex]$f(x) \leq 0$[/tex]



Answer :

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]