Certainly! Let's analyze the function [tex]\( f(x) = |x - 3| + 4 \)[/tex] step-by-step to determine its range.
1. Understanding the Absolute Value Function:
The function [tex]\( f(x) \)[/tex] includes an absolute value term, [tex]\( |x - 3| \)[/tex]. The absolute value function [tex]\( |y| \)[/tex] represents the distance of [tex]\( y \)[/tex] from zero on the number line, and is always non-negative. Therefore, [tex]\( |x - 3| \geq 0 \)[/tex] for all real numbers [tex]\( x \)[/tex].
2. Find the Minimum Value:
Since [tex]\( |x - 3| \geq 0 \)[/tex], the smallest value [tex]\( |x - 3| \)[/tex] can take is 0. This happens when [tex]\( x = 3 \)[/tex].
Substituting [tex]\( x = 3 \)[/tex] into the function:
[tex]\[
f(3) = |3 - 3| + 4 = 0 + 4 = 4
\][/tex]
Therefore, the minimum value of [tex]\( f(x) \)[/tex] is 4.
3. Determine How [tex]\( f(x) \)[/tex] Changes:
As [tex]\( x \)[/tex] moves away from 3, [tex]\( |x - 3| \)[/tex] increases. Because [tex]\( |x - 3| \)[/tex] can grow without bound as [tex]\( x \)[/tex] moves further from 3, [tex]\( f(x) \)[/tex] can also increase without bound. So, [tex]\( f(x) \)[/tex] can take any value greater than or equal to 4.
4. The Range:
Combining our findings, we see that [tex]\( f(x) \)[/tex] starts at 4 and can increase to any value greater than 4. Therefore, the range of [tex]\( f(x) \)[/tex] can be written as:
[tex]\[
R: \{ f(x) \in \mathbb{R} \mid f(x) \geq 4 \}
\][/tex]
Thus, the correct range for the function [tex]\( f(x) = |x - 3| + 4 \)[/tex] is:
[tex]\[
R: \{ f(x) \in \mathbb{R} \mid f(x) \geq 4 \}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{R: \{ f(x) \in \mathbb{R} \mid f(x) \geq 4 \}}
\][/tex]
