To determine the range of the function [tex]\( f(x) = |x| + 3 \)[/tex], let’s break it down step by step:
1. Analyze the Function:
- The function [tex]\( f(x) = |x| + 3 \)[/tex] involves the absolute value of [tex]\( x \)[/tex], denoted [tex]\( |x| \)[/tex], and a constant term [tex]\( +3 \)[/tex].
- The absolute value function [tex]\( |x| \)[/tex] returns the non-negative distance of [tex]\( x \)[/tex] from zero. This means [tex]\( |x| \geq 0 \)[/tex] for all [tex]\( x \)[/tex].
2. Evaluate the Minimum Value:
- The minimum value of [tex]\( |x| \)[/tex] is [tex]\( 0 \)[/tex]. This happens when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = |0| + 3 = 0 + 3 = 3.
\][/tex]
- Therefore, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 3 \)[/tex].
3. Evaluate Larger Values:
- As [tex]\( x \)[/tex] increases or decreases away from [tex]\( 0 \)[/tex], [tex]\( |x| \)[/tex] increases. Therefore, [tex]\( |x| + 3 \)[/tex] also increases.
- There is no upper bound to the value [tex]\( |x| \)[/tex] can achieve, hence [tex]\( f(x) \)[/tex] can take any value greater than or equal to [tex]\( 3 \)[/tex].
4. Determine the Range:
- From the observations, [tex]\( f(x) \)[/tex] can be as small as [tex]\( 3 \)[/tex] but can grow larger without any limit.
- Therefore, the range of [tex]\( f(x) \)[/tex] consists of all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq 3 \)[/tex].
Based on these considerations, the correct answer is:
[tex]\[
\{ y \mid 3 \leq y < \infty \}
\][/tex]
So, the range of [tex]\( f(x) = |x| + 3 \)[/tex] is given by the set:
[tex]\[
\{ y \mid 3 \leq y < \infty \}.
\][/tex]
Thus, the choice is:
[tex]\[
\boxed{\{ y \mid 3 \leq y < \infty \}}
\][/tex]
