To determine the range of the function [tex]\( f(x) = |x| - 3 \)[/tex], we start by understanding the behavior of the given formula.
1. The absolute value function [tex]\( |x| \)[/tex] always outputs a non-negative value, meaning [tex]\( |x| \)[/tex] is always [tex]\( \geq 0 \)[/tex] for any real number [tex]\( x \)[/tex].
2. This implies that the minimum value [tex]\( |x| \)[/tex] can attain is 0.
Now, evaluate [tex]\( f(x) = |x| - 3 \)[/tex] at this minimum value:
[tex]\[ f(0) = |0| - 3 = 0 - 3 = -3 \][/tex]
Thus, [tex]\( f(x) \geq -3 \)[/tex] is indeed correct.
Since [tex]\( |x| \)[/tex] can increase indefinitely as [tex]\( x \)[/tex] moves away from zero, the output of [tex]\( f(x) = |x| - 3 \)[/tex] will also increase without bound.
Therefore, the function can take any value starting from [tex]\(-3\)[/tex] and extending to positive infinity. In interval notation, this can be represented as:
[tex]\[ \text{Range: } [-3, \infty) \][/tex]
Thus, the correct answer is:
[tex]\[
\{ f(x) \in \mathbb{R} \mid f(x) \geq -3 \}
\][/tex]
