To determine the domain of the function [tex]\( h(x) = |x + 7| \)[/tex], we need to consider the nature of the absolute value function. The absolute value function, denoted as [tex]\( | \cdot | \)[/tex], is defined for all real numbers. This means that for any real number [tex]\( x \)[/tex], [tex]\( |x| \)[/tex] is well-defined and yields a non-negative real number.
Given [tex]\( h(x) = |x + 7| \)[/tex], let's analyze its domain step by step:
1. Identify the Expression Inside the Absolute Value:
- The expression inside the absolute value is [tex]\( x + 7 \)[/tex].
2. Determine the Set of All Possible Values for [tex]\( x \)[/tex]:
- The expression [tex]\( x + 7 \)[/tex] is a linear polynomial and is defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions like division by zero or taking the square root of a negative number.
3. Conclusion:
- Since the absolute value function [tex]\( | \cdot | \)[/tex] is defined for any real number and the expression [tex]\( x + 7 \)[/tex] is also valid for every real number [tex]\( x \)[/tex], the function [tex]\( h(x) = |x + 7| \)[/tex] is also defined for all real numbers.
- Therefore, the domain of [tex]\( h(x) = |x + 7| \)[/tex] is all real numbers.
So, the domain of [tex]\( h(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
Thus, the correct answer is:
B. [tex]\( (-\infty, \infty) \)[/tex]
