To determine the range of the function [tex]\( f(x) = |x-3| + 4 \)[/tex], we need to understand how the function behaves over the entire set of possible values for [tex]\( x \)[/tex].
1. Understanding the Absolute Value Function [tex]\( |x-3| \)[/tex]
- The expression [tex]\( |x-3| \)[/tex] represents the absolute value of [tex]\( x-3 \)[/tex], which is always non-negative. Specifically, for any value of [tex]\( x \)[/tex], [tex]\( |x-3| \geq 0 \)[/tex].
2. Behavior of [tex]\( |x-3| + 4 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( |x-3| = |3-3| = 0 \)[/tex]. Therefore, [tex]\( f(x) = 0 + 4 = 4 \)[/tex].
- For any other value of [tex]\( x \)[/tex], [tex]\( |x-3| \)[/tex] will be positive, meaning [tex]\( |x-3| > 0 \)[/tex]. Thus, [tex]\( f(x) = |x-3| + 4 > 4 \)[/tex].
3. Minimum Value of [tex]\( f(x) \)[/tex]
- Since the smallest value [tex]\( |x-3| \)[/tex] can take is 0, the minimum value of [tex]\( f(x) \)[/tex] is when [tex]\( |x-3| = 0 \)[/tex], giving us [tex]\( f(x) = 4 \)[/tex].
4. Maximum Value of [tex]\( f(x) \)[/tex]
- As [tex]\( x \)[/tex] moves away from 3, [tex]\( |x-3| \)[/tex] increases without bound. Therefore, [tex]\( f(x) = |x-3| + 4 \)[/tex] increases without bound. There is no upper limit to the value of [tex]\( f(x) \)[/tex], so it can increase towards infinity.
5. Conclusion on the Range of [tex]\( f(x) \)[/tex]
- Based on the behavior of [tex]\( f(x) \)[/tex], we can conclude that the range of the function starts at 4 and increases without any upper limit.
- Hence, the range of the function [tex]\( f(x) = |x-3| + 4 \)[/tex] is all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq 4 \)[/tex].
Therefore, the correct answer to the question about the range of [tex]\( f(x) = |x-3| + 4 \)[/tex] is:
[tex]\[ \text{Range: } \{ f(x) \in \mathbb{R} \mid f(x) \geq 4 \} \][/tex]
