To determine the domain and range of the function [tex]\( f(x) = |x + 6| \)[/tex], let's analyze each part step-by-step.
### Domain
The domain of a function refers to all possible input values (x-values) for which the function is defined.
For the function [tex]\( f(x) = |x + 6| \)[/tex]:
- The absolute value function [tex]\( | \cdot | \)[/tex] is defined for all real numbers.
- Therefore, there are no restrictions on the values of [tex]\( x \)[/tex] that can be input into the function.
Hence, the domain of [tex]\( f(x) = |x + 6| \)[/tex] is all real numbers, which we can write in interval notation as:
[tex]\[ (-\infty, \infty) \][/tex]
### Range
The range of a function refers to all possible output values (y-values) that the function can produce.
For the function [tex]\( f(x) = |x + 6| \)[/tex]:
- The expression [tex]\( |x + 6| \)[/tex] represents the absolute value of [tex]\( x + 6 \)[/tex], which means it will always be non-negative.
- Therefore, the output of this function will always be greater than or equal to 0 regardless of the value of [tex]\( x \)[/tex].
So, the range of [tex]\( f(x) = |x + 6| \)[/tex] is all non-negative real numbers, which we can write in interval notation as:
[tex]\[ [0, \infty) \][/tex]
### Conclusion
Combining these results, we have:
- The domain of [tex]\( f(x) = |x + 6| \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( f(x) = |x + 6| \)[/tex] is [tex]\( [0, \infty) \)[/tex].
Therefore, the correct set of answers for the given function is:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( [0, \infty) \)[/tex]
