To determine the range of the function [tex]\( f(x) = |x| - 5 \)[/tex], let's analyze the given function step-by-step:
1. Understanding the Absolute Value Function:
- The absolute value function, [tex]\( |x| \)[/tex], returns the non-negative value of [tex]\( x \)[/tex].
- Mathematically, if [tex]\( x \geq 0 \)[/tex], then [tex]\( |x| = x \)[/tex]. If [tex]\( x < 0 \)[/tex], then [tex]\( |x| = -x \)[/tex].
- Therefore, the range of [tex]\( |x| \)[/tex] is [tex]\([0, \infty)\)[/tex], since the absolute value of [tex]\( x \)[/tex] can never be negative.
2. Shifting the Function:
- The function [tex]\( f(x) = |x| - 5 \)[/tex] can be written as the absolute value function [tex]\( |x| \)[/tex] shifted downward by 5 units.
- Shifting a function downward by 5 units means subtracting 5 from every value of the function.
3. Determining the New Range:
- Since the range of [tex]\( |x| \)[/tex] is [tex]\([0, \infty)\)[/tex]:
- When we subtract 5 from each value in [tex]\([0, \infty)\)[/tex], the smallest value [tex]\( |x| \)[/tex] can take is 0.
- Subtracting 5 from 0 gives: [tex]\( 0 - 5 = -5 \)[/tex].
- Subtracting 5 from [tex]\(\infty\)[/tex] still gives us [tex]\(\infty\)[/tex].
- Thus, the new range of [tex]\( f(x) = |x| - 5 \)[/tex] is [tex]\([-5, \infty)\)[/tex].
4. Matching with Given Options:
- Looking at the provided options, the range [tex]\([-5, \infty)\)[/tex] is best represented by [tex]\( -5 \leq y < \infty \)[/tex].
Therefore, the correct answer is:
\[ \boxed{2} \
