To determine the range of the function [tex]\( f(x) = |x| + 5 \)[/tex], let's analyze the behavior of the function step-by-step.
1. Understand the Absolute Value Function:
- The absolute value function [tex]\( |x| \)[/tex] returns the non-negative value of [tex]\( x \)[/tex].
- For any real number [tex]\( x \)[/tex], [tex]\( |x| \geq 0 \)[/tex].
2. Evaluate the Function:
- The function [tex]\( f(x) = |x| + 5 \)[/tex] is the sum of [tex]\( |x| \)[/tex] and 5.
- Since [tex]\( |x| \geq 0 \)[/tex], we have [tex]\( f(x) = |x| + 5 \geq 5 \)[/tex].
3. Minimum Value:
- To find the minimum value of [tex]\( f(x) \)[/tex], consider the point where [tex]\( |x| = 0 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( |x| = 0 \)[/tex]. So, [tex]\( f(0) = 0 + 5 = 5 \)[/tex].
- Hence, [tex]\( f(x) \)[/tex] attains its minimum value of 5.
4. Range of [tex]\( f(x) \)[/tex]:
- Since [tex]\( f(x) = |x| + 5 \geq 5 \)[/tex], the function can take any value starting from 5 and increasing without bound.
- Hence, the range of [tex]\( f(x) \)[/tex] includes all real numbers greater than or equal to 5.
Based on the above analysis, the correct option from the given list is:
[tex]\[ R: \{ f(x) \in \mathbb{R} \mid f(x) \geq 5 \} \][/tex]
Therefore, the second option is the correct one.
