Consider this absolute value function:

[tex]\[ f(x) = |x + 3| \][/tex]

How can function [tex]\( f \)[/tex] be rewritten as a piecewise function?

[tex]\[
f(x) =
\begin{cases}
x + 3 & \text{if } x \geq -3 \\
-x - 3 & \text{if } x \ \textless \ -3
\end{cases}
\][/tex]



Answer :

To rewrite the absolute value function [tex]\( f(x) = |x+3| \)[/tex] as a piecewise function, you need to consider the behavior of the absolute value expression based on the value of [tex]\( x \)[/tex]. The absolute value function can be expressed as a piecewise function considering two cases:

1. When the expression inside the absolute value is non-negative, i.e., [tex]\( x + 3 \geq 0 \Rightarrow x \geq -3 \)[/tex].
2. When the expression inside the absolute value is negative, i.e., [tex]\( x + 3 < 0 \Rightarrow x < -3 \)[/tex].

For [tex]\( x \geq -3 \)[/tex], the absolute value function returns the expression itself:
[tex]\[ f(x) = x + 3 \][/tex]

For [tex]\( x < -3 \)[/tex], the absolute value function returns the negation of the expression:
[tex]\[ f(x) = -(x + 3) = -x - 3 \][/tex]

Putting it together, the piecewise function is:

[tex]\[ f(x) = \begin{cases} x + 3 & \text{if } x \geq -3 \\ -x - 3 & \text{if } x < -3 \end{cases} \][/tex]

So, the correct arrangement of the pieces is:
[tex]\[ f(x) = \begin{cases} x + 3 & \text{if } x \geq -3 \\ -x - 3 & \text{if } x < -3 \end{cases} \][/tex]