To determine the range of the function [tex]\( f(x) = |x - 3| + 4 \)[/tex], let’s analyze it step-by-step.
1. Understanding the Absolute Value Function:
- The expression [tex]\( |x - 3| \)[/tex] represents the absolute value of [tex]\( x - 3 \)[/tex].
- The absolute value function [tex]\( |x - 3| \)[/tex] is always non-negative, meaning [tex]\( |x - 3| \geq 0 \)[/tex].
2. Adding 4 to the Absolute Value:
- Since [tex]\( |x - 3| \geq 0 \)[/tex], if we add 4 to this expression, we get:
[tex]\[
|x - 3| + 4 \geq 0 + 4 = 4
\][/tex]
3. Finding the Minimum Value:
- The lowest value that [tex]\( f(x) = |x - 3| + 4 \)[/tex] can take is when [tex]\( |x - 3| = 0 \)[/tex]. This occurs when [tex]\( x = 3 \)[/tex].
- Thus, when [tex]\( x = 3 \)[/tex], we have:
[tex]\[
f(3) = |3 - 3| + 4 = 0 + 4 = 4
\][/tex]
4. Considering all Possible Values:
- For any other [tex]\( x \neq 3 \)[/tex], since [tex]\( |x - 3| > 0 \)[/tex], we have [tex]\( |x - 3| + 4 > 4 \)[/tex].
- Therefore, for any [tex]\( x \in \mathbb{R} \)[/tex], the function [tex]\( f(x) \)[/tex] will always be greater than or equal to 4.
5. Conclusion:
- The range of [tex]\( f(x) = |x - 3| + 4 \)[/tex] includes all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq 4 \)[/tex].
Thus, the correct answer is:
[tex]\[
R: \{f(x) \in \mathbb{R} \mid f(x) \geq 4\}
\][/tex]
