To determine the range of the function [tex]\( f(x) = |x| + 3 \)[/tex], let's analyze the function step-by-step.
1. Understanding the Absolute Value Function:
- The absolute value function [tex]\( |x| \)[/tex] is always non-negative. This means that [tex]\( |x| \geq 0 \)[/tex] for all [tex]\( x \in R \)[/tex].
2. Evaluating the Function:
- Since [tex]\( f(x) = |x| + 3 \)[/tex], we add 3 to the non-negative value of [tex]\( |x| \)[/tex].
- Therefore, [tex]\( f(x) \)[/tex] will always be at least [tex]\( 3 \)[/tex], because the smallest value [tex]\( |x| \)[/tex] can take is [tex]\( 0 \)[/tex].
3. Identifying the Minimum Value:
- The minimum value of [tex]\( f(x) \)[/tex] occurs when [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the function, we get:
[tex]\[
f(0) = |0| + 3 = 0 + 3 = 3
\][/tex]
4. Determining the Range:
- Since the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 3 \)[/tex] and the function increases as [tex]\( |x| \)[/tex] increases, the function takes all values starting from [tex]\( 3 \)[/tex] and extending to positive infinity.
- In symbolic form, we can say that [tex]\( f(x) \geq 3 \)[/tex].
Thus, the range of [tex]\( f(x) = |x| + 3 \)[/tex] is correctly described by:
[tex]\[
\{f(x) \in R \mid f(x) \geq 3\}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\{f(x) \in R \mid f(x) \geq 3\}
\][/tex]
