To determine the range of the function [tex]\( f(x) = |x + 5| \)[/tex], let's analyze the properties and behavior of the absolute value function.
1. Understanding the Absolute Value Function:
- The absolute value [tex]\( |y| \)[/tex] of a number [tex]\( y \)[/tex] is defined as the non-negative distance of [tex]\( y \)[/tex] from 0 on the number line. In other words, [tex]\( |y| \)[/tex] is always greater than or equal to zero.
- Because of this, the absolute value function [tex]\( f(x) = |x + 5| \)[/tex] will always output non-negative values.
2. Behavior of [tex]\( f(x) \)[/tex]:
- For any real number [tex]\( x \)[/tex], [tex]\( x + 5 \)[/tex] can take any value from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
- When you take the absolute value of [tex]\( x + 5 \)[/tex], regardless of whether [tex]\( x + 5 \)[/tex] was negative or positive, the output will be non-negative.
3. Finding the Range:
- Since the absolute value [tex]\( |x + 5| \)[/tex] always produces values that are zero or positive, the smallest value it can take is 0.
- As [tex]\( x \)[/tex] approaches [tex]\(-5\)[/tex], [tex]\( x + 5 \)[/tex] approaches 0, and thus [tex]\( |x + 5| \)[/tex] approaches 0.
- As [tex]\( x \)[/tex] takes on larger positive or negative values, [tex]\( |x + 5| \)[/tex] can become arbitrarily large, extending to infinity.
Therefore, the range of the function [tex]\( f(x) = |x + 5| \)[/tex] is all non-negative real numbers, which can be expressed in interval notation as:
[tex]\[
[0, \infty)
\][/tex]
In another form, the range is [tex]\((0, \text{oo})\)[/tex].
